Explanation:
Okay, so there's this figure skater who starts facing north, and she spins to her right. She spins a total of 2250 degrees. The question is asking which direction she ends up facing after that spin. Alright, let's break this down step by step.

First, I need to recall that when you spin around, each full rotation is 360 degrees. So, if she spins more than 360 degrees, she's making multiple full turns plus some extra degrees. Since she's spinning to her right, that means she's rotating clockwise, right? Because if you face north and turn right, you're going clockwise. Let me confirm that. If you're facing north and turn right, your right hand direction would be east, so yeah, clockwise. So, clockwise rotation.

So, the total angle is 2250 degrees. To figure out the direction, I need to find out how many full rotations she made and then the remaining degrees. Because after each full rotation (360 degrees), she ends up facing the same direction she started. So, the key is to find the equivalent angle between 0 and 360 degrees that is equivalent to 2250 degrees when considering rotations.

So, the first step is to divide 2250 by 360 to find out how many full rotations she made and the remainder. Let me do that calculation.

Let me compute 2250 divided by 360. Hmm. Let me think.

Well, 360 times 6 is 2160. Let me check: 360*6 = 2160. Then, 2250 - 2160 = 90. So, 2250 degrees is equal to 6 full rotations (which bring her back to the original direction) plus 90 degrees. Therefore, the total rotation is equivalent to 90 degrees clockwise from her starting direction.

Wait, but let me verify that division again. Let me do it step by step. Let's divide 2250 by 360. Let me write this as:

2250 ÷ 360. Let me simplify the numbers. Both numbers can be divided by 10 first: 225 ÷ 36. Let me divide numerator and denominator by 9: 225 ÷ 9 = 25, and 36 ÷ 9 = 4. So now, it's 25 ÷ 4. Which is 6.25. So that's 6 full rotations and 0.25 of a rotation. Since 0.25 of a rotation is 90 degrees (because 0.25 * 360 = 90). So that matches my previous calculation. Therefore, 2250 degrees is equivalent to 90 degrees of rotation.

Therefore, starting from north, if she spins 90 degrees clockwise, where will she end up?

Let me visualize. Facing north, turning right (clockwise) 90 degrees. So, north to east is 90 degrees. So, after 90 degrees, she is facing east. Wait, but hold on. Wait, if you face north and turn right (clockwise), your first 90 degrees would be east, then another 90 would be south, then another 90 west, then another 90 back to north. So, yes, 90 degrees clockwise from north is east.

But wait, hold on. Wait, let me check again. If you start facing north, and turn 90 degrees to the right, you are facing east. So, that seems correct.

So then, according to this calculation, she ends up facing east. But let me check again, because sometimes when you do these calculations, it's easy to make a mistake. Let me verify.

Alternatively, perhaps I can think of the total rotation as 2250 degrees clockwise. Let me compute how many full circles that is.

But since after every 360 degrees, it's the same direction, so subtract multiples of 360 until it's less than 360. So, 2250 divided by 360.

Alternatively, 2250 divided by 360: Let me compute 360*6=2160. 2250-2160=90. So 90 degrees. Therefore, same as before. So 90 degrees clockwise.

Therefore, starting at north, 90 degrees clockwise is east. Therefore, answer is east? But wait, hold on. Let me think again. Wait, is it 90 degrees? Wait, the problem says she spins 2250 degrees. So, maybe there's a mistake here?

Wait, 6 full rotations is 2160 degrees. 2250 - 2160 is 90, so 90 degrees. So, yes. So 90 degrees. So, facing north, turning 90 degrees clockwise is east.

But let me think again: maybe I should check with another method. For example, think in terms of modulo 360. So, 2250 mod 360. Let me compute that.

2250 divided by 360 is 6 with a remainder of 90. So, 2250 mod 360 is 90. Therefore, the effective rotation is 90 degrees. Therefore, direction is north + 90 degrees clockwise.

But let me verify with another approach. Let's think about how many 90-degree turns are in 2250 degrees. Each 90 degrees is a quarter turn. So 2250 divided by 90 is 25. So, 25 quarter turns. Each quarter turn is a 90-degree clockwise turn. So, 25 quarter turns. Now, each full rotation is 4 quarter turns. So, 25 divided by 4 is 6 with a remainder of 1. So, 6 full rotations (which brings her back to original direction) plus 1 quarter turn. Therefore, 1 quarter turn clockwise from north is east. Therefore, same conclusion. So, direction is east.

Wait, so seems like answer is east. But let me check once again, because maybe I made a mistake in the direction?

Wait, if you start facing north, and turn right (clockwise) 90 degrees, you end up facing east. Then, another 90 degrees (total 180) would be south, another 90 (total 270) west, another 90 (total 360) back to north. So, yes. So 90 degrees is east.

But let me check with another way. Let me think about the total rotation. If she spins 2250 degrees, which is 6 full spins (2160) plus 90. So, after 6 full spins, she is facing north again. Then she spins an additional 90 degrees. So, from north, 90 degrees clockwise is east. Therefore, answer is east. Therefore, I think the answer is east.

Wait, but let me check once more. Maybe the problem says "spins to her right" so maybe she is spinning around her own axis? But the direction is determined by the rotation. So, if she starts facing north, and spins to her right, meaning that she's rotating clockwise. So, yes. Therefore, 90 degrees clockwise from north is east. So answer should be east.

But wait, hold on. Wait, maybe there's a mistake in the calculation? Because 2250 degrees is a lot. Let me think. Let me do the division again. 360*6=2160, 2250-2160=90. So, 90 degrees. That's correct. But maybe I need to consider that when you spin to your right, it's a clockwise rotation. But maybe I need to think about the direction again?

Alternatively, maybe I should think in terms of coordinates. Let me assign coordinates. Let me imagine that the skater is at the origin, facing north. Let me think of directions as compass directions.

North is 0 degrees, east is 90 degrees, south is 180, west is 270, and back to north at 360. So, if she spins clockwise, her direction increases by the angle. Wait, but if she spins clockwise, then her direction would be measured as increasing clockwise? Wait, maybe in terms of standard position, angles are measured counterclockwise from the positive x-axis, but here, maybe we need to adjust.

But perhaps another way: if you are facing north, and you turn clockwise (to your right) by an angle, then your new direction can be calculated by adding the angle to your current direction, modulo 360. However, since we are dealing with directions, perhaps we need to map the angles properly.

But let me think. Let's define the initial direction as 0 degrees (north). Then, turning clockwise by θ degrees would result in a direction of (0 + θ) mod 360 degrees. However, in standard compass terms, clockwise rotation would correspond to increasing angles. Wait, perhaps it's better to think that if you turn clockwise, then the direction is measured as follows:

Starting at north (0 degrees). Turning 90 degrees clockwise (right) would point east (90 degrees). Then another 90 degrees would be south (180), then west (270), then back to north (360). So yes, if you add the angle of rotation to the initial direction, modulo 360. So, in this case, the total rotation is 2250 degrees. So, the final direction is (0 + 2250) mod 360 degrees. Which is 2250 mod 360.

As calculated earlier, 2250 divided by 360 is 6*360=2160, remainder 90. So, 90 degrees. Therefore, direction is 90 degrees, which is east. Therefore, the answer is east.

But hold on, maybe there's confusion with the direction of rotation? Let me check again. Suppose the skater is facing north. If she spins to her right, which is clockwise. So, if you imagine facing north, turning right (clockwise) 90 degrees, you face east. If you turn another 90, south, etc. So, yes, adding degrees to the initial direction.

Alternatively, if you consider the standard mathematical convention where angles are measured counterclockwise from the positive x-axis (east), then north would be 90 degrees, east is 0 degrees, south is 270, etc. But in this case, the problem is using compass directions, so maybe we need to adjust accordingly. But regardless, the key is that the rotation is clockwise, so adding degrees to the initial direction.

Alternatively, if we consider that in the standard position (mathematical angle), turning clockwise would be negative angles. But since we are dealing with compass directions, perhaps it's better to stick with the compass direction.

Alternatively, let me think of it as:

Starting direction: North.

Each 90 degrees clockwise: East, South, West, North...

So, the number of 90-degree increments in 2250 degrees is 2250 / 90 = 25. So, 25 increments of 90 degrees. Each full circle is 4 increments. Therefore, 25 divided by 4 is 6 with a remainder of 1. Therefore, 6 full circles (which brings her back to North) plus 1 increment of 90 degrees. Therefore, North + 90 degrees clockwise = East. So again, same conclusion. Therefore, answer is East.

Wait, but maybe I made a mistake in the remainder? Let me check 25 divided by 4. 4*6=24. So, 25-24=1. So, remainder 1. So yes, 1 increment. So, 1*90=90 degrees. So, same result.

Alternatively, perhaps I need to think of her spinning 2250 degrees to the right, which is equivalent to spinning 2250 degrees clockwise. Therefore, starting at North, the direction is North + 2250 degrees clockwise. Since each 360 degrees brings her back, subtract 360 as many times as needed.

Alternatively, another way to think: 2250 divided by 360. Let me compute 360*6=2160. 2250-2160=90. So, 90 degrees. So, same.

Another way: if you have 2250 degrees, how many times does 360 go into it?

Well, 360*5=1800. 2250-1800=450. 450-360=90. So total is 5+1=6 full rotations, remainder 90. Same as before.

So, conclusion is 90 degrees. So, starting from North, 90 degrees clockwise is East. Therefore, the answer is East.

But wait, hold on. Wait, I have a doubt here. Because if you spin 2250 degrees, which is 6 full circles (2160) plus 90 degrees. So, 6 full circles brings her back to North. Then, spinning an additional 90 degrees to the right (clockwise) would face her East. So, answer is East.

But let me check with an example. Suppose she spins 90 degrees, she faces East. 180 degrees, South. 270 degrees, West. 360 degrees, back to North. 450 degrees, same as 90 degrees, East. So, 2250 degrees is 6*360 + 90. So, same as 90 degrees. So, yes, East.

But just to make sure, maybe I should think of another way. Let me think in terms of vectors or something. Suppose her initial direction is north. Let me assign coordinates. Let me think of her position as (0,0), facing north, which is the positive y-axis. If she spins clockwise, her direction vector will rotate clockwise.

But perhaps this is overcomplicating. But maybe for verification.

If she spins 90 degrees clockwise, her direction is east, which is the positive x-axis. So, the direction vector would be (1,0). If she spins another 90 degrees, direction is south, direction vector (0,-1). Then another 90, west, (-1,0). Another 90, back to north. So, yes, each 90 degrees is a quarter turn clockwise.

Therefore, 2250 degrees is 6*360 + 90, so same as 90 degrees. Therefore, direction is east.

Wait, but maybe there's something wrong here? Because 2250 seems like a very large number. But since rotations repeat every 360, it's okay. So, 2250 divided by 360 is 6.25. So, 6 full turns and a quarter turn. So, yeah, quarter turn is 90 degrees. So, answer is east.

Alternatively, maybe I need to check with modulo operation. Let me compute 2250 mod 360. Let me compute 360*6=2160. 2250-2160=90. Therefore, 2250 mod 360 is 90. So, 90 degrees. Therefore, direction is east.

Therefore, after all these checks, I think the answer is east.

Wait, but hold on. I just want to be absolutely sure. Because sometimes problems can have nuances. For example, if the skater was spinning around her own axis, but the direction is determined by her facing. So, if she spins 2250 degrees, then her facing direction depends on the total rotation. Since each full rotation brings her back to the original direction, so the key is the remainder.

Alternatively, maybe there's a mistake in assuming that 2250 degrees is equivalent to 90 degrees. Let me check with another example. Suppose she spins 360 degrees, she ends up facing north. 720 degrees, same. 2250 degrees, same as 2250 - 6*360 = 2250 - 2160 = 90. So, yes, same as 90 degrees. So, she ends up facing east.

Alternatively, maybe I should think in terms of right-hand rule or something? No, because it's a simple rotation in 2D. The right-hand rule is more for 3D rotations, but here it's just a planar rotation.

Alternatively, maybe I should think about the direction as a number. Let me assign numbers to directions:

North = 0 degrees

East = 90 degrees

South = 180 degrees

West = 270 degrees

Then, rotating clockwise adds degrees. So, starting at 0 degrees (North). After rotating θ degrees clockwise, the direction is (0 + θ) mod 360. Therefore, for θ = 2250, direction is 2250 mod 360 = 90 degrees, which is East. Therefore, answer is East.

Yes, so all methods lead to East.

But let me check again with the initial problem statement. It says she spins to her right. So, if she is facing north, spinning to her right would be clockwise. So, yes. So, adding degrees. Therefore, answer is East.

Wait, but I have a lingering doubt because sometimes when people talk about directions, they might have different conventions. For example, maybe in some contexts, turning right is considered counterclockwise? But no, that doesn't make sense. If you are facing north, turning right (clockwise) would be towards east, which is standard. Turning left (counterclockwise) would be towards west.

Therefore, there's no confusion here.

Alternatively, let me check with an example. Suppose she spins 180 degrees to the right. Starting at north, spinning 180 degrees to the right would face her south. Which is correct. Similarly, spinning 270 degrees to the right would face her west. Which is correct. Therefore, spinning 90 degrees to the right faces east. So, 2250 degrees is equivalent to 90 degrees, so east.

Therefore, I think the answer is East. So, the final answer is East, which is \boxed{East}.

Wait, but let me check one more time. Maybe I miscalculated the remainder?

Let me do 2250 divided by 360. So 360*6 is 2160. Subtract that from 2250: 2250 - 2160 = 90. Yes, so remainder 90. Therefore, 2250 degrees is equivalent to 90 degrees. So, yes. So, answer is East. So, I think this is correct. I can't find any mistake in the reasoning. So, I think the answer is East.

**Final Answer**
\boxed{East}

To determine the final direction of the figure skater after spinning **2250 degrees to her right**, we need to analyze how much of that rotation results in a net directional change.

---

### Step 1: Understand the Rotation Direction

- Since the skater is facing **north** at the start and **spins to her right**, this means she is rotating **clockwise**.
- A full rotation (360 degrees) brings her back to the same orientation.

---

### Step 2: Reduce the Total Rotation Using Modulo

To find the **effective rotation**, we reduce 2250 degrees modulo 360, as any full rotation (360 degrees) brings her back to her original direction.

$$
2250 \div 360 = 6 \text{ remainder } 90
$$

Thus:

$$
2250 \equiv 90 \ (\text{mod } 360)
$$

This tells us that **2250 degrees clockwise** is **equivalent to 90 degrees clockwise** from her original direction.

---

### Step 3: Determine the Final Direction

Starting from **north**, a **90-degree clockwise rotation** results in her facing **east**:

- North → East (90° clockwise)
- East → South (another 90°)
- South → West (another 90°)
- West → North (another 90°)

Therefore, **90 degrees clockwise from north** = **east**.

---

### Final Answer

$$
\boxed{East}
$$

[Assistant]
Explanation:
Okay, so there's this figure skater who starts facing north, and she spins to her right. She spins a total of 2250 degrees. The question is asking which direction she ends up facing after that spin. Alright, let's break this down step by step.

First, I need to recall that when you spin around, each full rotation is 360 degrees. So, if she spins more than 360 degrees, she's making multiple full turns plus some extra degrees. Since she's spinning to her right, that means she's rotating clockwise, right? Because if you face north and turn right, you're going clockwise. Let me confirm that. If you're facing north and turn right, your right hand direction would be east, so yeah, clockwise. So, clockwise rotation.

So, the total angle is 2250 degrees. To figure out the direction, I need to find out how many full rotations she made and then the remaining degrees. Because after each full rotation (360 degrees), she ends up facing the same direction she started. So, the key is to find the equivalent angle between 0 and 360 degrees that is equivalent to 2250 degrees when considering rotations.

So, the first step is to divide 2250 by 360 to find out how many full rotations she made and the remainder. Let me do that calculation.

Let me compute 2250 divided by 360. Hmm. Let me think.

Well, 360 times 6 is 2160. Let me check: 360*6 = 2160. Then, 2250 - 2160 = 90. So, 2250 degrees is equal to 6 full rotations (which bring her back to the original direction) plus 90 degrees. Therefore, the total rotation is equivalent to 90 degrees clockwise from her starting direction.

Wait, but let me verify that division again. Let me do it step by step. Let's divide 2250 by 360. Let me write this as:

2250 ÷ 360. Let me simplify the numbers. Both numbers can be divided by 10 first: 225 ÷ 36. Let me divide numerator and denominator by 9: 225 ÷ 9 = 25, and 36 ÷ 9 = 4. So now, it's 25 ÷ 4. Which is 6.25. So that's 6 full rotations and 0.25 of a rotation. Since 0.25 of a rotation is 90 degrees (because 0.25 * 360 = 90). So that matches my previous calculation. Therefore, 2250 degrees is equivalent to 90 degrees of rotation.

Therefore, starting from north, if she spins 90 degrees clockwise, where will she end up?

Let me visualize. Facing north, turning right (clockwise) 90 degrees. So, north to east is 90 degrees. So, after 90 degrees, she is facing east. Wait, but hold on. Wait, if you face north and turn right (clockwise), your first 90 degrees would be east, then another 90 would be south, then another 90 west, then another 90 back to north. So, yes, 90 degrees clockwise from north is east.

But wait, hold on. Wait, let me check again. If you start facing north, and turn 90 degrees to the right, you are facing east. So, that seems correct.

So then, according to this calculation, she ends up facing east. But let me check again, because sometimes when you do these calculations, it's easy to make a mistake. Let me verify.

Alternatively, perhaps I can think of the total rotation as 2250 degrees clockwise. Let me compute how many full circles that is.

But since after every 360 degrees, it's the same direction, so subtract multiples of 360 until it's less than 360. So, 2250 divided by 360.

Alternatively, 2250 divided by 360: Let me compute 360*6=2160. 2250-2160=90. So 90 degrees. Therefore, same as before. So 90 degrees clockwise.

Therefore, starting at north, 90 degrees clockwise is east. Therefore, answer is east? But wait, hold on. Let me think again. Wait, is it 90 degrees? Wait, the problem says she spins 2250 degrees. So, maybe there's a mistake here?

Wait, 6 full rotations is 2160 degrees. 2250 - 2160 is 90, so 90 degrees. So, yes. So 90 degrees. So, facing north, turning 90 degrees clockwise is east.

But let me think again: maybe I should check with another method. For example, think in terms of modulo 360. So, 2250 mod 360. Let me compute that.

2250 divided by 360 is 6 with a remainder of 90. So, 2250 mod 360 is 90. Therefore, the effective rotation is 90 degrees. Therefore, direction is north + 90 degrees clockwise.

But let me verify with another approach. Let's think about how many 90-degree turns are in 2250 degrees. Each 90 degrees is a quarter turn. So 2250 divided by 90 is 25. So, 25 quarter turns. Each quarter turn is a 90-degree clockwise turn. So, 25 quarter turns. Now, each full rotation is 4 quarter turns. So, 25 divided by 4 is 6 with a remainder of 1. So, 6 full rotations (which brings her back to original direction) plus 1 quarter turn. Therefore, 1 quarter turn clockwise from north is east. Therefore, same conclusion. So, direction is east.

Wait, so seems like answer is east. But let me check once again, because maybe I made a mistake in the direction?

Wait, if you start facing north, and turn right (clockwise) 90 degrees, you end up facing east. Then, another 90 degrees (total 180) would be south, another 90 (total 270) west, another 90 (total 360) back to north. So, yes. So 90 degrees is east.

But let me check with another way. Let me think about the total rotation. If she spins 2250 degrees, which is 6 full spins (2160) plus 90. So, after 6 full spins, she is facing north again. Then she spins an additional 90 degrees. So, from north, 90 degrees clockwise is east. Therefore, answer is east. Therefore, I think the answer is east.

Wait, but let me check once more. Maybe the problem says "spins to her right" so maybe she is spinning around her own axis? But the direction is determined by the rotation. So, if she starts facing north, and spins to her right, meaning that she's rotating clockwise. So, yes. Therefore, 90 degrees clockwise from north is east. So answer should be east.

But wait, hold on. Wait, maybe there's a mistake in the calculation? Because 2250 degrees is a lot. Let me think. Let me do the division again. 360*6=2160, 2250-2160=90. So, 90 degrees. That's correct. But maybe I need to consider that when you spin to your right, it's a clockwise rotation. But maybe I need to think about the direction again?

Alternatively, maybe I should think in terms of coordinates. Let me assign coordinates. Let me imagine that the skater is at the origin, facing north. Let me think of directions as compass directions.

North is 0 degrees, east is 90 degrees, south is 180, west is 270, and back to north at 360. So, if she spins clockwise, her direction increases by the angle. Wait, but if she spins clockwise, then her direction would be measured as increasing clockwise? Wait, maybe in terms of standard position, angles are measured counterclockwise from the positive x-axis, but here, maybe we need to adjust.

But perhaps another way: if you are facing north, and you turn clockwise (to your right) by an angle, then your new direction can be calculated by adding the angle to your current direction, modulo 360. However, since we are dealing with directions, perhaps we need to map the angles properly.

But let me think. Let's define the initial direction as 0 degrees (north). Then, turning clockwise by θ degrees would result in a direction of (0 + θ) mod 360 degrees. However, in standard compass terms, clockwise rotation would correspond to increasing angles. Wait, perhaps it's better to think that if you turn clockwise, then the direction is measured as follows:

Starting at north (0 degrees). Turning 90 degrees clockwise (right) would point east (90 degrees). Then another 90 degrees would be south (180), then west (270), then back to north (360). So yes, if you add the angle of rotation to the initial direction, modulo 360. So, in this case, the total rotation is 2250 degrees. So, the final direction is (0 + 2250) mod 360 degrees. Which is 2250 mod 360.

As calculated earlier, 2250 divided by 360 is 6*360=2160, remainder 90. So, 90 degrees. Therefore, direction is 90 degrees, which is east. Therefore, the answer is east.

But hold on, maybe there's confusion with the direction of rotation? Let me check again. Suppose the skater is facing north. If she spins to her right, which is clockwise. So, if you imagine facing north, turning right (clockwise) 90 degrees, you face east. If you turn another 90, south, etc. So, yes, adding degrees to the initial direction.

Alternatively, if you consider the standard mathematical convention where angles are measured counterclockwise from the positive x-axis (east), then north would be 90 degrees, east is 0 degrees, south is 270, etc. But in this case, the problem is using compass directions, so maybe we need to adjust accordingly. But regardless, the key is that the rotation is clockwise, so adding degrees to the initial direction.

Alternatively, if we consider that in the standard position (mathematical angle), turning clockwise would be negative angles. But since we are dealing with compass directions, perhaps it's better to stick with the compass direction.

Alternatively, let me think of it as:

Starting direction: North.

Each 90 degrees clockwise: East, South, West, North...

So, the number of 90-degree increments in 2250 degrees is 2250 / 90 = 25. So, 25 increments of 90 degrees. Each full circle is 4 increments. Therefore, 25 divided by 4 is 6 with a remainder of 1. Therefore, 6 full circles (which brings her back to North) plus 1 increment of 90 degrees. Therefore, North + 90 degrees clockwise = East. So again, same conclusion. Therefore, answer is East.

Wait, but maybe I made a mistake in the remainder? Let me check 25 divided by 4. 4*6=24. So, 25-24=1. So, remainder 1. So yes, 1 increment. So, 1*90=90 degrees. So, same result.

Alternatively, perhaps I need to think of her spinning 2250 degrees to the right, which is equivalent to spinning 2250 degrees clockwise. Therefore, starting at North, the direction is North + 2250 degrees clockwise. Since each 360 degrees brings her back, subtract 360 as many times as needed.

Alternatively, another way to think: 2250 divided by 360. Let me compute 360*6=2160. 2250-2160=90. So, 90 degrees. So, same.

Another way: if you have 2250 degrees, how many times does 360 go into it?

Well, 360*5=1800. 2250-1800=450. 450-360=90. So total is 5+1=6 full rotations, remainder 90. Same as before.

So, conclusion is 90 degrees. So, starting from North, 90 degrees clockwise is East. Therefore, the answer is East.

But wait, hold on. Wait, I have a doubt here. Because if you spin 2250 degrees, which is 6 full circles (2160) plus 90 degrees. So, 6 full circles brings her back to North. Then, spinning an additional 90 degrees to the right (clockwise) would face her East. So, answer is East.

But let me check with an example. Suppose she spins 90 degrees, she faces East. 180 degrees, South. 270 degrees, West. 360 degrees, back to North. 450 degrees, same as 90 degrees, East. So, 2250 degrees is 6*360 + 90. So, same as 90 degrees. So, yes, East.

But just to make sure, maybe I should think of another way. Let me think in terms of vectors or something. Suppose her initial direction is north. Let me assign coordinates. Let me think of her position as (0,0), facing north, which is the positive y-axis. If she spins clockwise, her direction vector will rotate clockwise.

But perhaps this is overcomplicating. But maybe for verification.

If she spins 90 degrees clockwise, her direction is east, which is the positive x-axis. So, the direction vector would be (1,0). If she spins another 90 degrees, direction is south, direction vector (0,-1). Then another 90, west, (-1,0). Another 90, back to north. So, yes, each 90 degrees is a quarter turn clockwise.

Therefore, 2250 degrees is 6*360 + 90, so same as 90 degrees. Therefore, direction is east.

Wait, but maybe there's something wrong here? Because 2250 seems like a very large number. But since rotations repeat every 360, it's okay. So, 2250 divided by 360 is 6.25. So, 6 full turns and a quarter turn. So, yeah, quarter turn is 90 degrees. So, answer is east.

Alternatively, maybe I need to check with modulo operation. Let me compute 2250 mod 360. Let me compute 360*6=2160. 2250-2160=90. Therefore, 2250 mod 360 is 90. So, 90 degrees. Therefore, direction is east.

Therefore, after all these checks, I think the answer is east.

Wait, but hold on. I just want to be absolutely sure. Because sometimes problems can have nuances. For example, if the skater was spinning around her own axis, but the direction is determined by her facing. So, if she spins 2250 degrees, then her facing direction depends on the total rotation. Since each full rotation brings her back to the original direction, so the key is the remainder.

Alternatively, maybe there's a mistake in assuming that 2250 degrees is equivalent to 90 degrees. Let me check with another example. Suppose she spins 360 degrees, she ends up facing north. 720 degrees, same. 2250 degrees, same as 2250 - 6*360 = 2250 - 2160 = 90. So, yes, same as 90 degrees. So, she ends up facing east.

Alternatively, maybe I should think in terms of right-hand rule or something? No, because it's a simple rotation in 2D. The right-hand rule is more for 3D rotations, but here it's just a planar rotation.

Alternatively, maybe I should think about the direction as a number. Let me assign numbers to directions:

North = 0 degrees

East = 90 degrees

South = 180 degrees

West = 270 degrees

Then, rotating clockwise adds degrees. So, starting at 0 degrees (North). After rotating θ degrees clockwise, the direction is (0 + θ) mod 360. Therefore, for θ = 2250, direction is 2250 mod 360 = 90 degrees, which is East. Therefore, answer is East.

Yes, so all methods lead to East.

But let me check again with the initial problem statement. It says she spins to her right. So, if she is facing north, spinning to her right would be clockwise. So, yes. So, adding degrees. Therefore, answer is East.

Wait, but I have a lingering doubt because sometimes when people talk about directions, they might have different conventions. For example, maybe in some contexts, turning right is considered counterclockwise? But no, that doesn't make sense. If you are facing north, turning right (clockwise) would be towards east, which is standard. Turning left (counterclockwise) would be towards west.

Therefore, there's no confusion here.

Alternatively, let me check with an example. Suppose she spins 180 degrees to the right. Starting at north, spinning 180 degrees to the right would face her south. Which is correct. Similarly, spinning 270 degrees to the right would face her west. Which is correct. Therefore, spinning 90 degrees to the right faces east. So, 2250 degrees is equivalent to 90 degrees, so east.

Therefore, I think the answer is East. So, the final answer is East, which is \boxed{East}.

Wait, but let me check one more time. Maybe I miscalculated the remainder?

Let me do 2250 divided by 360. So 360*6 is 2160. Subtract that from 2250: 2250 - 2160 = 90. Yes, so remainder 90. Therefore, 2250 degrees is equivalent to 90 degrees. So, yes. So, answer is East. So, I think this is correct. I can't find any mistake in the reasoning. So, I think the answer is East.

**Final Answer**
\boxed{East}

To determine the final direction of the figure skater after spinning **2250 degrees to her right**, we need to analyze how much of that rotation results in a net directional change.

---

### Step 1: Understand the Rotation Direction

- Since the skater is facing **north** at the start and **spins to her right**, this means she is rotating **clockwise**.
- A full rotation (360 degrees) brings her back to the same orientation.

---

### Step 2: Reduce the Total Rotation Using Modulo

To find the **effective rotation**, we reduce 2250 degrees modulo 360, as any full rotation (360 degrees) brings her back to her original direction.

$$
2250 \div 360 = 6 \text{ remainder } 90
$$

Thus:

$$
2250 \equiv 90 \ (\text{mod } 360)
$$

This tells us that **2250 degrees clockwise** is **equivalent to 90 degrees clockwise** from her original direction.

---

### Step 3: Determine the Final Direction

Starting from **north**, a **90-degree clockwise rotation** results in her facing **east**:

- North → East (90° clockwise)
- East → South (another 90°)
- South → West (another 90°)
- West → North (another 90°)

Therefore, **90 degrees clockwise from north** = **east**.

---

### Final Answer

$$
\boxed{East}
$$

Solution: Each full circle is 360 degrees.  Dividing 360 into 2250 gives a quotient of 6 with a remainder of 90.  So, she spins 90 degrees to her right past north, which leaves her facing $\boxed{\text{east}}$.


Explanation:
Okay, so I need to factor the expression ab + 5b + 2a + 10. Let me think about how to approach this. Factoring expressions usually involves finding common factors or grouping terms in a way that allows me to factor them out. Let me start by looking at each term individually to see if there are any common factors.

First, the terms are ab, 5b, 2a, and 10. Let me check for common factors across all terms. The variables involved are a, b, and constants. If I look at the coefficients: 1 (for ab), 5 (for 5b), 2 (for 2a), and 10 (constant term). The greatest common divisor of 1, 5, 2, and 10 is 1, so there's no common numerical factor across all terms. However, maybe there are common factors in pairs of terms.

Looking at the first two terms: ab and 5b. Both have a 'b' in them. So I can factor out a 'b' from those two terms. Let me try that:

ab + 5b = b(a + 5)

Okay, so that gives me b(a + 5). Now, the remaining terms are 2a and 10. Let me check if those have a common factor. The coefficients are 2 and 10, so their greatest common divisor is 2. So factoring out a 2 from 2a + 10 gives me 2(a + 5).

Wait a second, so now I have:

b(a + 5) + 2(a + 5)

Oh! Both terms now have a common factor of (a + 5). That's great! So I can factor out (a + 5) from both terms. Let me write that down:

(a + 5)(b + 2)

Let me check if that's correct by expanding it back out. Let's multiply (a + 5)(b + 2):

First, a times b is ab.

Then, a times 2 is 2a.

Then, 5 times b is 5b.

Finally, 5 times 2 is 10.

Adding those up: ab + 2a + 5b + 10. Which is the same as the original expression ab + 5b + 2a + 10. The order of the terms is different, but since addition is commutative, it's the same expression. Therefore, the factoring is correct.

So, the factored form is (a + 5)(b + 2).

But let me just make sure I didn't miss any other possible ways to factor this. For example, could there be another grouping? Let me try grouping the terms differently. Suppose I group ab + 2a and 5b + 10. Let's see:

ab + 2a = a(b + 2)

5b + 10 = 5(b + 2)

Then, combining those two results gives a(b + 2) + 5(b + 2) = (a + 5)(b + 2). Same result.

So regardless of how I group them, I end up with the same factors. That gives me more confidence that this is the correct factorization.

Alternatively, if I tried to factor by some other method, like looking for a common binomial factor or something else, but I don't think there's another way here. The key was recognizing that after grouping the first two terms and the last two terms, or alternatively grouping ab with 2a and 5b with 10, you can factor out common binomials.

Another way to check if the factoring is complete is to see if the factors can be factored further. In this case, (a + 5) and (b + 2) are both linear terms and cannot be factored further, so this is the complete factorization.

Let me also think if there's a possibility that the original expression could be factored in a different way, but I don't think so. The original expression is a quadratic in terms of a and b, but since it's a two-variable polynomial, factoring into two binomials is the standard approach here.

Another check: if I plug in some numbers for a and b, the original expression and the factored form should give the same result. Let me choose a = 1, b = 1.

Original expression: ab + 5b + 2a + 10 = (1)(1) + 5(1) + 2(1) + 10 = 1 + 5 + 2 + 10 = 18.

Factored form: (a + 5)(b + 2) = (1 + 5)(1 + 2) = (6)(3) = 18. Same result.

Let me try another set: a = 2, b = 3.

Original expression: (2)(3) + 5(3) + 2(2) + 10 = 6 + 15 + 4 + 10 = 35.

Factored form: (2 + 5)(3 + 2) = (7)(5) = 35. Also matches.

Another check: a = 0, b = 0.

Original: 0 + 0 + 0 + 10 = 10.

Factored: (0 + 5)(0 + 2) = 5 * 2 = 10. Correct again.

So, these checks confirm that my factoring is correct. Therefore, the expression factors into (a + 5)(b + 2).

**Final Answer**
The factored form of the expression is \boxed{(a + 5)(b + 2)}.

To factor the expression:

$$
ab + 5b + 2a + 10
$$

---

### Step 1: Group Terms Strategically

We aim to identify common factors that can be factored out from groups of terms. A common and effective technique is **grouping**.

Group the first two terms and the last two terms:

$$
(ab + 5b) + (2a + 10)
$$

---

### Step 2: Factor Out Common Factors from Each Group

From the first group:
$$
ab + 5b = b(a + 5)
$$

From the second group:
$$
2a + 10 = 2(a + 5)
$$

Now the expression becomes:

$$
b(a + 5) + 2(a + 5)
$$

---

### Step 3: Factor Out the Common Binomial

Both terms now share a common factor of $ (a + 5) $. We can factor that out:

$$
(a + 5)(b + 2)
$$

---

### Step 4: Verify the Factoring

We can double-check by expanding the factored form:

$$
(a + 5)(b + 2) = ab + 2a + 5b + 10
$$

Which matches the original expression. Thus, our factorization is correct.

---

### Final Answer

$$
\boxed{(a + 5)(b + 2)}
$$

[Assistant]
Explanation:
Okay, so I need to factor the expression ab + 5b + 2a + 10. Let me think about how to approach this. Factoring expressions usually involves finding common factors or grouping terms in a way that allows me to factor them out. Let me start by looking at each term individually to see if there are any common factors.

First, the terms are ab, 5b, 2a, and 10. Let me check for common factors across all terms. The variables involved are a, b, and constants. If I look at the coefficients: 1 (for ab), 5 (for 5b), 2 (for 2a), and 10 (constant term). The greatest common divisor of 1, 5, 2, and 10 is 1, so there's no common numerical factor across all terms. However, maybe there are common factors in pairs of terms.

Looking at the first two terms: ab and 5b. Both have a 'b' in them. So I can factor out a 'b' from those two terms. Let me try that:

ab + 5b = b(a + 5)

Okay, so that gives me b(a + 5). Now, the remaining terms are 2a and 10. Let me check if those have a common factor. The coefficients are 2 and 10, so their greatest common divisor is 2. So factoring out a 2 from 2a + 10 gives me 2(a + 5).

Wait a second, so now I have:

b(a + 5) + 2(a + 5)

Oh! Both terms now have a common factor of (a + 5). That's great! So I can factor out (a + 5) from both terms. Let me write that down:

(a + 5)(b + 2)

Let me check if that's correct by expanding it back out. Let's multiply (a + 5)(b + 2):

First, a times b is ab.

Then, a times 2 is 2a.

Then, 5 times b is 5b.

Finally, 5 times 2 is 10.

Adding those up: ab + 2a + 5b + 10. Which is the same as the original expression ab + 5b + 2a + 10. The order of the terms is different, but since addition is commutative, it's the same expression. Therefore, the factoring is correct.

So, the factored form is (a + 5)(b + 2).

But let me just make sure I didn't miss any other possible ways to factor this. For example, could there be another grouping? Let me try grouping the terms differently. Suppose I group ab + 2a and 5b + 10. Let's see:

ab + 2a = a(b + 2)

5b + 10 = 5(b + 2)

Then, combining those two results gives a(b + 2) + 5(b + 2) = (a + 5)(b + 2). Same result.

So regardless of how I group them, I end up with the same factors. That gives me more confidence that this is the correct factorization.

Alternatively, if I tried to factor by some other method, like looking for a common binomial factor or something else, but I don't think there's another way here. The key was recognizing that after grouping the first two terms and the last two terms, or alternatively grouping ab with 2a and 5b with 10, you can factor out common binomials.

Another way to check if the factoring is complete is to see if the factors can be factored further. In this case, (a + 5) and (b + 2) are both linear terms and cannot be factored further, so this is the complete factorization.

Let me also think if there's a possibility that the original expression could be factored in a different way, but I don't think so. The original expression is a quadratic in terms of a and b, but since it's a two-variable polynomial, factoring into two binomials is the standard approach here.

Another check: if I plug in some numbers for a and b, the original expression and the factored form should give the same result. Let me choose a = 1, b = 1.

Original expression: ab + 5b + 2a + 10 = (1)(1) + 5(1) + 2(1) + 10 = 1 + 5 + 2 + 10 = 18.

Factored form: (a + 5)(b + 2) = (1 + 5)(1 + 2) = (6)(3) = 18. Same result.

Let me try another set: a = 2, b = 3.

Original expression: (2)(3) + 5(3) + 2(2) + 10 = 6 + 15 + 4 + 10 = 35.

Factored form: (2 + 5)(3 + 2) = (7)(5) = 35. Also matches.

Another check: a = 0, b = 0.

Original: 0 + 0 + 0 + 10 = 10.

Factored: (0 + 5)(0 + 2) = 5 * 2 = 10. Correct again.

So, these checks confirm that my factoring is correct. Therefore, the expression factors into (a + 5)(b + 2).

**Final Answer**
The factored form of the expression is \boxed{(a + 5)(b + 2)}.

To factor the expression:

$$
ab + 5b + 2a + 10
$$

---

### Step 1: Group Terms Strategically

We aim to identify common factors that can be factored out from groups of terms. A common and effective technique is **grouping**.

Group the first two terms and the last two terms:

$$
(ab + 5b) + (2a + 10)
$$

---

### Step 2: Factor Out Common Factors from Each Group

From the first group:
$$
ab + 5b = b(a + 5)
$$

From the second group:
$$
2a + 10 = 2(a + 5)
$$

Now the expression becomes:

$$
b(a + 5) + 2(a + 5)
$$

---

### Step 3: Factor Out the Common Binomial

Both terms now share a common factor of $ (a + 5) $. We can factor that out:

$$
(a + 5)(b + 2)
$$

---

### Step 4: Verify the Factoring

We can double-check by expanding the factored form:

$$
(a + 5)(b + 2) = ab + 2a + 5b + 10
$$

Which matches the original expression. Thus, our factorization is correct.

---

### Final Answer

$$
\boxed{(a + 5)(b + 2)}
$$

Solution: We have $ab +5b+2a+10 = ab + 5b+2a + 2\cdot 5$, so we have a straightforward application of Simon's Favorite Factoring Trick: \[ab + 5b+2a+10 = \boxed{(a+5)(b+2)}.\]


Explanation:
Okay, let's try to figure out this problem. So, Zach has three bags and a bunch of pencils. He needs to place the greatest number of pencils possible into each of the three bags, but he also has to keep the number of pencils in each bag the same. The question is asking for the greatest number of pencils he could have left over. Hmm, let me parse that again.

Wait, maybe the problem is saying that he wants to divide the pencils into three bags such that each bag has the same number of pencils, but he wants to maximize the number of pencils in each bag. But then, there's some leftover pencils? Wait, but if he's trying to place the greatest number of pencils possible into each bag while keeping the number in each bag the same, maybe it's about dividing the pencils into three bags with equal numbers, and wanting to maximize the number in each bag. But then, if there are leftovers, how does that work?

Wait, maybe the problem is similar to division with remainder. Let me think. If you have a certain number of pencils, and you want to divide them into three bags with the same number in each, the remainder would be the leftover pencils. But the question says he wants to place the greatest number of pencils possible into each bag, while keeping the number in each bag the same. So maybe it's like, he wants to maximize the number of pencils per bag, but there's some leftover? But how does that make sense? If you maximize the number per bag, you would minimize the leftover, right? Wait, maybe the problem is phrased differently.

Wait, perhaps the problem is that he has to divide the pencils into three bags, each with the same number of pencils, and he wants the number in each bag to be as large as possible. However, there might be some leftover pencils that can't be distributed. But then, the question is asking for the greatest number of pencils he could have left over. Wait, but if he is trying to put as many as possible into each bag, then the leftover pencils would be the remainder when the total number is divided by 3. But how can we determine the maximum possible leftover? Since when you divide by 3, the maximum remainder is 2. So is the answer 2?

But that seems too straightforward. Maybe I need to check the problem again. Let me re-read it.

"Zach has three bags and a bunch of pencils to be placed into the bags. He is told to place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same. What is the greatest number of pencils he could have left over?"

Wait, maybe the problem is not about dividing them equally but about having the same number in each bag? But if he wants the number in each bag to be the same, then the leftover would be the remainder. However, the problem says "the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same." Wait, this is confusing. Maybe the problem is that he has to put the same number in each bag, but he wants to maximize the number in each bag. That would mean that he wants to divide all the pencils equally into three bags, with no leftovers. Because if there are leftovers, then he can't put the same number in each bag. Wait, but if he can't put the same number in each bag, then the number of pencils he can put into each bag is floor(total / 3), and the leftover is total mod 3. So if he wants to maximize the number of pencils in each bag, he would have to have total divisible by 3, so that there are no leftovers. But the question is asking for the greatest number of pencils he could have left over. So maybe the problem is phrased differently.

Alternatively, maybe the problem is that the number of pencils in each bag must be the same, but he wants to maximize the number of pencils in each bag. However, there might be some constraints on the number of pencils he has? Wait, but the problem says "a bunch of pencils," so maybe the number of pencils is arbitrary. But the question is asking for the maximum possible leftover pencils. If the number of pencils is arbitrary, then the maximum number of leftovers when dividing by 3 is 2. But why is the question asking for that?

Alternatively, perhaps there is a misunderstanding in the problem's phrasing. Maybe it's not that he wants to divide them equally, but he wants to place pencils into the bags such that each bag has the same number of pencils, and he wants the number in each bag to be as large as possible. However, there might be some constraints?

Wait, let me think again. If you have some number of pencils, and you want to put them into three bags with the same number in each, then the number in each bag is floor(total / 3), and the remainder is total mod 3. However, if the problem wants the greatest number of pencils he could have left over, meaning that he wants to maximize the remainder when dividing by 3. But since the remainder when dividing by 3 can be at most 2, then the answer is 2. However, maybe the problem is in another context? For example, if there's some other constraint?

Wait, but maybe the problem is not about dividing all pencils into the bags, but placing the greatest number of pencils possible into each bag, but keeping the number in each bag the same. Maybe the total number of pencils is not fixed? Wait, the problem says "a bunch of pencils to be placed into the bags." So maybe the number of pencils is variable, and he wants to choose the number of pencils such that when divided into three bags, each bag has the same number of pencils, and he wants to maximize the number in each bag. But if he can choose the number of pencils, then he can have any number. Wait, but he wants the greatest number of pencils possible into each bag, so maybe the number is unbounded? That doesn't make sense.

Alternatively, perhaps there's a misinterpretation here. Maybe the problem is similar to the classic problem where you have to divide pencils into bags with certain conditions. Wait, maybe there is a typo or mistranslation? Let me try to think again.

Alternatively, maybe the problem is that he has to place pencils into three bags such that each bag has the same number of pencils, but he wants to maximize the number in each bag, but there's some condition that the number of pencils he has is not fixed. However, he wants to know the maximum number of pencils that could be left over. Wait, but left over from what? If he is trying to put as many as possible into each bag, then there's no leftover. If there is a leftover, that would imply that he couldn't put all pencils into the bags. Maybe the problem is that he has to distribute the pencils into three bags, but the number in each bag must be the same, and he wants to maximize the number in each bag. However, if he has more pencils than can be divided evenly, then some are left over. But the problem says "the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same." Maybe this is a translation issue.

Wait, maybe the problem is in Chinese or another language and translated to English? The original might have a different nuance. But assuming the translation is accurate, perhaps the key is that he has to place pencils into the bags, with each bag having the same number, and he wants to maximize the number per bag. But if he can choose the number of pencils, then he can make them all be divisible by 3, so there is no leftover. But maybe the problem is asking about the maximum possible leftover given that he wants to place as many as possible into each bag? That is, if he wants to maximize the number per bag, but there's a limit on the total number of pencils?

But the problem says "a bunch of pencils," which is vague. Wait, maybe the problem is similar to the one where you have to distribute pencils into bags with certain conditions, and the leftover is the maximum possible. For example, if you have to divide pencils into three bags with the same number in each, and you want the maximum number of pencils left over, meaning that the total number is not a multiple of three. But then, the maximum possible leftover is two. But why would the problem be asking for that? It seems too simple.

Wait, but maybe there's another interpretation. Maybe the problem is saying that he wants to place the greatest number of pencils possible into each bag, but he also needs to keep the number of pencils in each bag the same. Wait, maybe he has to have the same number of pencils in each bag, and he wants to know the maximum number of pencils that could be left over. But again, if he wants to place as many as possible into each bag, then the leftover would be minimized. So this is confusing.

Alternatively, maybe the problem is that he has three bags and wants to put pencils into them such that each bag has the same number of pencils as the others. But he wants to maximize the number in each bag, but there's some constraint that he only has a certain number of pencils. However, since the problem doesn't specify the total number of pencils, maybe the question is about the maximum possible remainder when the total number of pencils is divided by 3, which is 2. But then why mention "the greatest number of pencils possible into each of the three bags"? That might be redundant.

Alternatively, maybe the problem is about distributing pencils into three bags with the same number in each, but also considering some other constraints. For example, maybe each bag has to have a certain number of pencils, but the problem is not clear.

Wait, let's check similar problems. For example, if you have a certain number of items and you want to divide them into groups, the remainder is the leftover. If you want to maximize the remainder, it's the maximum possible remainder when divided by the number of groups. Since the number of groups is 3, the maximum remainder is 2. Therefore, the answer is 2. But maybe there's more to it.

Alternatively, maybe the problem is about dividing pencils into three bags, each with the same number, but the number of pencils in each bag must be as large as possible, but there's a condition that the total number of pencils is not a multiple of three. However, without knowing the total number, it's impossible to determine the remainder. Therefore, the problem must be asking for the maximum possible remainder, which is 2.

Alternatively, maybe the problem is in the context of some other constraints. For example, if the number of pencils is to be distributed such that each bag has the same number, but there's some other condition like each bag must have at least one pencil. But even so, the maximum remainder is still 2.

Wait, maybe the question is phrased as "the greatest number of pencils he could have left over" meaning that the leftover pencils are the ones that couldn't be distributed equally, and he wants to know the maximum possible number of such leftovers. Since when dividing by three, the maximum remainder is 2, then the answer is 2. Therefore, maybe the answer is \boxed{2}?

But let me think again. Suppose that the problem is from a math competition or something, and the answer is different. Wait, could there be another interpretation?

Suppose that Zach has three bags and wants to put pencils into them. He wants to place the greatest number of pencils possible into each bag while keeping the number of pencils in each bag the same. Wait, perhaps "keeping the number of pencils in each bag the same" as some previous condition? Maybe there was a prior condition that the number of pencils in each bag must be the same, and he wants to maximize that number. But if that's the case, then the number of pencils in each bag would be floor(total / 3), and the leftover is total - 3 * floor(total / 3). But again, the maximum possible leftover is 2.

Alternatively, maybe the problem is about the number of pencils per bag being the same, but the total number of pencils is fixed, and he's trying to maximize the number per bag. However, if the total is fixed, then the number per bag is total / 3, and there's no leftover. But again, unless there is a constraint on the total number.

Wait, maybe the problem is saying that he has to place pencils into the bags such that each bag has the same number of pencils, and he wants to maximize the number of pencils in each bag, but there's some other condition. For example, maybe the number of pencils in each bag must be a certain type of number, like a prime or something. But since the problem doesn't specify, that seems unlikely.

Alternatively, maybe the problem is referring to the fact that if you have three bags, and you want to put as many pencils as possible into each, but you have to keep the number of pencils in each bag the same. So, for example, if you want each bag to have the same number of pencils, and you want to maximize that number, then again, you need the total number of pencils to be divisible by 3, so there's no leftover. But if it's not divisible by 3, then you can't have all bags with the same number. Therefore, the maximum number of pencils that can be placed into each bag is floor(total / 3), and the leftover is total - 3 * floor(total / 3). But since the problem says "the greatest number of pencils he could have left over," maybe the question is about the maximum possible value of that leftover. Since if you choose the total number of pencils to be such that the remainder is maximized, which is 2, then the answer is 2.

Therefore, maybe the answer is 2. But I'm not sure why the problem is phrased the way it is. Maybe there's a trick here. Wait, let's think of specific numbers.

Suppose the total number of pencils is N. If N is divisible by 3, then there are 0 leftovers. If N = 3k + 1, then the leftovers are 1. If N = 3k + 2, leftovers are 2. So the maximum possible leftover is 2. Therefore, regardless of N, the maximum number of pencils that could be left over is 2. So the answer is 2.

But maybe the problem is different. For example, if there's some other condition. Suppose that the number of pencils in each bag must be the same, and Zach has to put pencils into the bags, but he can also have some pencils not in the bags. Wait, but the problem says "a bunch of pencils to be placed into the bags." So all pencils must be placed into the bags? Or some can be left over?

Wait, the problem says "place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same." So perhaps he is allowed to have some pencils left over, but he wants to maximize the number in each bag. In that case, the number in each bag is floor(N / 3), and the left over is N - 3 * floor(N / 3). But since he wants the greatest number of pencils possible into each bag, which would be achieved when N is as large as possible, but since N is not specified, the problem is asking for the maximum possible remainder, which is 2.

Alternatively, maybe the problem is a trick question. If you have to keep the number of pencils in each bag the same, then you can't have any left over, because if you have leftovers, then the number in each bag isn't the same. Wait, that seems contradictory. If you have leftovers, that would mean that the number in each bag is not the same. If you want the number in each bag to be the same, then you have to distribute all pencils into the bags, so there are no leftovers. Therefore, if he is told to place the greatest number of pencils possible into each bag while keeping the number in each bag the same, he must distribute all pencils equally, so there are no leftovers. But then the question says "the greatest number of pencils he could have left over," which would be zero. But that contradicts previous thoughts.

This is confusing. Which interpretation is correct? Let's re-examine the exact wording:

"Zach has three bags and a bunch of pencils to be placed into the bags. He is told to place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same. What is the greatest number of pencils he could have left over?"

So the key here is that he is told to place the greatest number of pencils possible into each bag while keeping the number in each bag the same. So, he wants to maximize the number in each bag, which would mean making each bag as full as possible. But if you want the number in each bag to be the same, then you have to divide the total number of pencils by 3. If there's a remainder, you can't have the same number in each bag. So if you want to maximize the number in each bag, you would take the floor division, which would leave a remainder. However, if you are allowed to have a remainder, then you can't have the same number in each bag. Hence, to keep the number in each bag the same, you have to distribute all pencils, hence no remainder. But that seems conflicting.

Wait, maybe the problem is that he has to have the same number of pencils in each bag, and he wants to put the greatest number possible into each bag, but he is allowed to have some pencils left over. Wait, but if he is allowed to have pencils left over, then he can't have the same number in each bag. For example, suppose he has 7 pencils. If he wants to put the same number in each of three bags, the maximum he can put in each is 2 (since 3*2=6, leaving 1 leftover). So in this case, the number of pencils left over is 1. But if he had 8 pencils, he can put 2 in each bag, leaving 2. So the maximum leftover is 2.

But the problem says "place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same." So it's like, he wants to maximize the number per bag, but also keep them equal. Therefore, the number per bag is floor(N / 3), and the leftover is N - 3 * floor(N / 3). Since the question is asking for the greatest number of pencils he could have left over, which is the maximum possible remainder, which is 2. Hence, the answer is 2.

But why does the wording say "place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same"? It seems like he is trying to maximize the number per bag, but also ensuring that they are the same. Therefore, the number per bag is floor(N / 3), and the leftover is the remainder. Hence, the maximum possible leftover is 2. Therefore, the answer is 2.

Alternatively, if the problem had said that he has to divide all pencils into the bags with equal numbers, then there would be no leftovers. But since it says "place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same," it implies that he's not required to place all pencils, but wants to place as many as possible into the bags with equal numbers. Hence, the rest are leftovers. Therefore, the maximum possible leftover is 2.

So I think the answer is 2. Therefore, \boxed{2}

**Final Answer**
\boxed{2}

To solve the problem, we need to carefully interpret the phrasing:

> "Zach has three bags and a bunch of pencils to be placed into the bags. He is told to place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same. What is the greatest number of pencils he could have left over?"

---

### Step-by-Step Interpretation

The key parts of the problem are:

- **Three bags**: There are exactly three bags.
- **A bunch of pencils**: The total number of pencils is unspecified but is a large, unknown number.
- **Place the greatest number of pencils possible into each of the three bags**: This implies he wants to maximize the number of pencils per bag.
- **While also keeping the number of pencils in each bag the same**: So, each bag must contain the **same number** of pencils.
- **What is the greatest number of pencils he could have left over?**

---

### Mathematical Framework

Suppose the total number of pencils is **N**. If Zach wants to place the maximum number of pencils into each of the three bags while keeping the number in each bag the same (i.e., **equal** per bag), he must divide the pencils as evenly as possible across the three bags.

So, if he can place **k** pencils in each of the three bags, that uses up **3k** pencils. The **remaining** pencils that cannot be divided equally will be:

$$
\text{Leftover} = N - 3k
$$

To maximize the number of pencils that could be left over, we must **maximize the size of the remainder** when **N** is divided by 3.

---

### Arithmetic Insight

When dividing a number by 3, the **maximum possible remainder** is **2**. This occurs when the total number of pencils is one less than a multiple of 3, e.g., 7, 10, 13, etc.

In such cases:
- $ N = 3k + 2 $
- The number of pencils he can place into each bag (while keeping them equal) is **k**
- The number left over is **2**

Any higher remainder (3 or more) is not possible since dividing by 3 yields remainders only in the range **[0, 2]**.

---

### Conclusion

Since the maximum possible **remainder** when dividing any number of pencils by 3 is **2**, this is also the **greatest number of pencils he could have left over**, given the constraints.

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, let's try to figure out this problem. So, Zach has three bags and a bunch of pencils. He needs to place the greatest number of pencils possible into each of the three bags, but he also has to keep the number of pencils in each bag the same. The question is asking for the greatest number of pencils he could have left over. Hmm, let me parse that again.

Wait, maybe the problem is saying that he wants to divide the pencils into three bags such that each bag has the same number of pencils, but he wants to maximize the number of pencils in each bag. But then, there's some leftover pencils? Wait, but if he's trying to place the greatest number of pencils possible into each bag while keeping the number in each bag the same, maybe it's about dividing the pencils into three bags with equal numbers, and wanting to maximize the number in each bag. But then, if there are leftovers, how does that work?

Wait, maybe the problem is similar to division with remainder. Let me think. If you have a certain number of pencils, and you want to divide them into three bags with the same number in each, the remainder would be the leftover pencils. But the question says he wants to place the greatest number of pencils possible into each bag, while keeping the number in each bag the same. So maybe it's like, he wants to maximize the number of pencils per bag, but there's some leftover? But how does that make sense? If you maximize the number per bag, you would minimize the leftover, right? Wait, maybe the problem is phrased differently.

Wait, perhaps the problem is that he has to divide the pencils into three bags, each with the same number of pencils, and he wants the number in each bag to be as large as possible. However, there might be some leftover pencils that can't be distributed. But then, the question is asking for the greatest number of pencils he could have left over. Wait, but if he is trying to put as many as possible into each bag, then the leftover pencils would be the remainder when the total number is divided by 3. But how can we determine the maximum possible leftover? Since when you divide by 3, the maximum remainder is 2. So is the answer 2?

But that seems too straightforward. Maybe I need to check the problem again. Let me re-read it.

"Zach has three bags and a bunch of pencils to be placed into the bags. He is told to place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same. What is the greatest number of pencils he could have left over?"

Wait, maybe the problem is not about dividing them equally but about having the same number in each bag? But if he wants the number in each bag to be the same, then the leftover would be the remainder. However, the problem says "the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same." Wait, this is confusing. Maybe the problem is that he has to put the same number in each bag, but he wants to maximize the number in each bag. That would mean that he wants to divide all the pencils equally into three bags, with no leftovers. Because if there are leftovers, then he can't put the same number in each bag. Wait, but if he can't put the same number in each bag, then the number of pencils he can put into each bag is floor(total / 3), and the leftover is total mod 3. So if he wants to maximize the number of pencils in each bag, he would have to have total divisible by 3, so that there are no leftovers. But the question is asking for the greatest number of pencils he could have left over. So maybe the problem is phrased differently.

Alternatively, maybe the problem is that the number of pencils in each bag must be the same, but he wants to maximize the number of pencils in each bag. However, there might be some constraints on the number of pencils he has? Wait, but the problem says "a bunch of pencils," so maybe the number of pencils is arbitrary. But the question is asking for the maximum possible leftover pencils. If the number of pencils is arbitrary, then the maximum number of leftovers when dividing by 3 is 2. But why is the question asking for that?

Alternatively, perhaps there is a misunderstanding in the problem's phrasing. Maybe it's not that he wants to divide them equally, but he wants to place pencils into the bags such that each bag has the same number of pencils, and he wants the number in each bag to be as large as possible. However, there might be some constraints?

Wait, let me think again. If you have some number of pencils, and you want to put them into three bags with the same number in each, then the number in each bag is floor(total / 3), and the remainder is total mod 3. However, if the problem wants the greatest number of pencils he could have left over, meaning that he wants to maximize the remainder when dividing by 3. But since the remainder when dividing by 3 can be at most 2, then the answer is 2. However, maybe the problem is in another context? For example, if there's some other constraint?

Wait, but maybe the problem is not about dividing all pencils into the bags, but placing the greatest number of pencils possible into each bag, but keeping the number in each bag the same. Maybe the total number of pencils is not fixed? Wait, the problem says "a bunch of pencils to be placed into the bags." So maybe the number of pencils is variable, and he wants to choose the number of pencils such that when divided into three bags, each bag has the same number of pencils, and he wants to maximize the number in each bag. But if he can choose the number of pencils, then he can have any number. Wait, but he wants the greatest number of pencils possible into each bag, so maybe the number is unbounded? That doesn't make sense.

Alternatively, perhaps there's a misinterpretation here. Maybe the problem is similar to the classic problem where you have to divide pencils into bags with certain conditions. Wait, maybe there is a typo or mistranslation? Let me try to think again.

Alternatively, maybe the problem is that he has to place pencils into three bags such that each bag has the same number of pencils, but he wants to maximize the number in each bag, but there's some condition that the number of pencils he has is not fixed. However, he wants to know the maximum number of pencils that could be left over. Wait, but left over from what? If he is trying to put as many as possible into each bag, then there's no leftover. If there is a leftover, that would imply that he couldn't put all pencils into the bags. Maybe the problem is that he has to distribute the pencils into three bags, but the number in each bag must be the same, and he wants to maximize the number in each bag. However, if he has more pencils than can be divided evenly, then some are left over. But the problem says "the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same." Maybe this is a translation issue.

Wait, maybe the problem is in Chinese or another language and translated to English? The original might have a different nuance. But assuming the translation is accurate, perhaps the key is that he has to place pencils into the bags, with each bag having the same number, and he wants to maximize the number per bag. But if he can choose the number of pencils, then he can make them all be divisible by 3, so there is no leftover. But maybe the problem is asking about the maximum possible leftover given that he wants to place as many as possible into each bag? That is, if he wants to maximize the number per bag, but there's a limit on the total number of pencils?

But the problem says "a bunch of pencils," which is vague. Wait, maybe the problem is similar to the one where you have to distribute pencils into bags with certain conditions, and the leftover is the maximum possible. For example, if you have to divide pencils into three bags with the same number in each, and you want the maximum number of pencils left over, meaning that the total number is not a multiple of three. But then, the maximum possible leftover is two. But why would the problem be asking for that? It seems too simple.

Wait, but maybe there's another interpretation. Maybe the problem is saying that he wants to place the greatest number of pencils possible into each bag, but he also needs to keep the number of pencils in each bag the same. Wait, maybe he has to have the same number of pencils in each bag, and he wants to know the maximum number of pencils that could be left over. But again, if he wants to place as many as possible into each bag, then the leftover would be minimized. So this is confusing.

Alternatively, maybe the problem is that he has three bags and wants to put pencils into them such that each bag has the same number of pencils as the others. But he wants to maximize the number in each bag, but there's some constraint that he only has a certain number of pencils. However, since the problem doesn't specify the total number of pencils, maybe the question is about the maximum possible remainder when the total number of pencils is divided by 3, which is 2. But then why mention "the greatest number of pencils possible into each of the three bags"? That might be redundant.

Alternatively, maybe the problem is about distributing pencils into three bags with the same number in each, but also considering some other constraints. For example, maybe each bag has to have a certain number of pencils, but the problem is not clear.

Wait, let's check similar problems. For example, if you have a certain number of items and you want to divide them into groups, the remainder is the leftover. If you want to maximize the remainder, it's the maximum possible remainder when divided by the number of groups. Since the number of groups is 3, the maximum remainder is 2. Therefore, the answer is 2. But maybe there's more to it.

Alternatively, maybe the problem is about dividing pencils into three bags, each with the same number, but the number of pencils in each bag must be as large as possible, but there's a condition that the total number of pencils is not a multiple of three. However, without knowing the total number, it's impossible to determine the remainder. Therefore, the problem must be asking for the maximum possible remainder, which is 2.

Alternatively, maybe the problem is in the context of some other constraints. For example, if the number of pencils is to be distributed such that each bag has the same number, but there's some other condition like each bag must have at least one pencil. But even so, the maximum remainder is still 2.

Wait, maybe the question is phrased as "the greatest number of pencils he could have left over" meaning that the leftover pencils are the ones that couldn't be distributed equally, and he wants to know the maximum possible number of such leftovers. Since when dividing by three, the maximum remainder is 2, then the answer is 2. Therefore, maybe the answer is \boxed{2}?

But let me think again. Suppose that the problem is from a math competition or something, and the answer is different. Wait, could there be another interpretation?

Suppose that Zach has three bags and wants to put pencils into them. He wants to place the greatest number of pencils possible into each bag while keeping the number of pencils in each bag the same. Wait, perhaps "keeping the number of pencils in each bag the same" as some previous condition? Maybe there was a prior condition that the number of pencils in each bag must be the same, and he wants to maximize that number. But if that's the case, then the number of pencils in each bag would be floor(total / 3), and the leftover is total - 3 * floor(total / 3). But again, the maximum possible leftover is 2.

Alternatively, maybe the problem is about the number of pencils per bag being the same, but the total number of pencils is fixed, and he's trying to maximize the number per bag. However, if the total is fixed, then the number per bag is total / 3, and there's no leftover. But again, unless there is a constraint on the total number.

Wait, maybe the problem is saying that he has to place pencils into the bags such that each bag has the same number of pencils, and he wants to maximize the number of pencils in each bag, but there's some other condition. For example, maybe the number of pencils in each bag must be a certain type of number, like a prime or something. But since the problem doesn't specify, that seems unlikely.

Alternatively, maybe the problem is referring to the fact that if you have three bags, and you want to put as many pencils as possible into each, but you have to keep the number of pencils in each bag the same. So, for example, if you want each bag to have the same number of pencils, and you want to maximize that number, then again, you need the total number of pencils to be divisible by 3, so there's no leftover. But if it's not divisible by 3, then you can't have all bags with the same number. Therefore, the maximum number of pencils that can be placed into each bag is floor(total / 3), and the leftover is total - 3 * floor(total / 3). But since the problem says "the greatest number of pencils he could have left over," maybe the question is about the maximum possible value of that leftover. Since if you choose the total number of pencils to be such that the remainder is maximized, which is 2, then the answer is 2.

Therefore, maybe the answer is 2. But I'm not sure why the problem is phrased the way it is. Maybe there's a trick here. Wait, let's think of specific numbers.

Suppose the total number of pencils is N. If N is divisible by 3, then there are 0 leftovers. If N = 3k + 1, then the leftovers are 1. If N = 3k + 2, leftovers are 2. So the maximum possible leftover is 2. Therefore, regardless of N, the maximum number of pencils that could be left over is 2. So the answer is 2.

But maybe the problem is different. For example, if there's some other condition. Suppose that the number of pencils in each bag must be the same, and Zach has to put pencils into the bags, but he can also have some pencils not in the bags. Wait, but the problem says "a bunch of pencils to be placed into the bags." So all pencils must be placed into the bags? Or some can be left over?

Wait, the problem says "place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same." So perhaps he is allowed to have some pencils left over, but he wants to maximize the number in each bag. In that case, the number in each bag is floor(N / 3), and the left over is N - 3 * floor(N / 3). But since he wants the greatest number of pencils possible into each bag, which would be achieved when N is as large as possible, but since N is not specified, the problem is asking for the maximum possible remainder, which is 2.

Alternatively, maybe the problem is a trick question. If you have to keep the number of pencils in each bag the same, then you can't have any left over, because if you have leftovers, then the number in each bag isn't the same. Wait, that seems contradictory. If you have leftovers, that would mean that the number in each bag is not the same. If you want the number in each bag to be the same, then you have to distribute all pencils into the bags, so there are no leftovers. Therefore, if he is told to place the greatest number of pencils possible into each bag while keeping the number in each bag the same, he must distribute all pencils equally, so there are no leftovers. But then the question says "the greatest number of pencils he could have left over," which would be zero. But that contradicts previous thoughts.

This is confusing. Which interpretation is correct? Let's re-examine the exact wording:

"Zach has three bags and a bunch of pencils to be placed into the bags. He is told to place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same. What is the greatest number of pencils he could have left over?"

So the key here is that he is told to place the greatest number of pencils possible into each bag while keeping the number in each bag the same. So, he wants to maximize the number in each bag, which would mean making each bag as full as possible. But if you want the number in each bag to be the same, then you have to divide the total number of pencils by 3. If there's a remainder, you can't have the same number in each bag. So if you want to maximize the number in each bag, you would take the floor division, which would leave a remainder. However, if you are allowed to have a remainder, then you can't have the same number in each bag. Hence, to keep the number in each bag the same, you have to distribute all pencils, hence no remainder. But that seems conflicting.

Wait, maybe the problem is that he has to have the same number of pencils in each bag, and he wants to put the greatest number possible into each bag, but he is allowed to have some pencils left over. Wait, but if he is allowed to have pencils left over, then he can't have the same number in each bag. For example, suppose he has 7 pencils. If he wants to put the same number in each of three bags, the maximum he can put in each is 2 (since 3*2=6, leaving 1 leftover). So in this case, the number of pencils left over is 1. But if he had 8 pencils, he can put 2 in each bag, leaving 2. So the maximum leftover is 2.

But the problem says "place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same." So it's like, he wants to maximize the number per bag, but also keep them equal. Therefore, the number per bag is floor(N / 3), and the leftover is N - 3 * floor(N / 3). Since the question is asking for the greatest number of pencils he could have left over, which is the maximum possible remainder, which is 2. Hence, the answer is 2.

But why does the wording say "place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same"? It seems like he is trying to maximize the number per bag, but also ensuring that they are the same. Therefore, the number per bag is floor(N / 3), and the leftover is the remainder. Hence, the maximum possible leftover is 2. Therefore, the answer is 2.

Alternatively, if the problem had said that he has to divide all pencils into the bags with equal numbers, then there would be no leftovers. But since it says "place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same," it implies that he's not required to place all pencils, but wants to place as many as possible into the bags with equal numbers. Hence, the rest are leftovers. Therefore, the maximum possible leftover is 2.

So I think the answer is 2. Therefore, \boxed{2}

**Final Answer**
\boxed{2}

To solve the problem, we need to carefully interpret the phrasing:

> "Zach has three bags and a bunch of pencils to be placed into the bags. He is told to place the greatest number of pencils possible into each of the three bags while also keeping the number of pencils in each bag the same. What is the greatest number of pencils he could have left over?"

---

### Step-by-Step Interpretation

The key parts of the problem are:

- **Three bags**: There are exactly three bags.
- **A bunch of pencils**: The total number of pencils is unspecified but is a large, unknown number.
- **Place the greatest number of pencils possible into each of the three bags**: This implies he wants to maximize the number of pencils per bag.
- **While also keeping the number of pencils in each bag the same**: So, each bag must contain the **same number** of pencils.
- **What is the greatest number of pencils he could have left over?**

---

### Mathematical Framework

Suppose the total number of pencils is **N**. If Zach wants to place the maximum number of pencils into each of the three bags while keeping the number in each bag the same (i.e., **equal** per bag), he must divide the pencils as evenly as possible across the three bags.

So, if he can place **k** pencils in each of the three bags, that uses up **3k** pencils. The **remaining** pencils that cannot be divided equally will be:

$$
\text{Leftover} = N - 3k
$$

To maximize the number of pencils that could be left over, we must **maximize the size of the remainder** when **N** is divided by 3.

---

### Arithmetic Insight

When dividing a number by 3, the **maximum possible remainder** is **2**. This occurs when the total number of pencils is one less than a multiple of 3, e.g., 7, 10, 13, etc.

In such cases:
- $ N = 3k + 2 $
- The number of pencils he can place into each bag (while keeping them equal) is **k**
- The number left over is **2**

Any higher remainder (3 or more) is not possible since dividing by 3 yields remainders only in the range **[0, 2]**.

---

### Conclusion

Since the maximum possible **remainder** when dividing any number of pencils by 3 is **2**, this is also the **greatest number of pencils he could have left over**, given the constraints.

$$
\boxed{2}
$$

Solution: If Zach has three or more pencils left over, then he can add another pencil to each bag.  Therefore, Zach can have at most $\boxed{2}$ pencils left over.


Explanation:
Okay, so I need to find the point on the curve given by the parameterization (x, y) = (t³ + 7, -3t² - 6t - 5) when t = 2. Hmm, let me think.

First, parameterized curves are basically curves where both x and y are expressed in terms of a parameter, which in this case is t. So, to find the point at t = 2, I just need to substitute t = 2 into both the x and y equations, right?

Let me write that down. For x, the equation is x = t³ + 7. So plugging in t = 2, that would be x = (2)³ + 7. Let me calculate that. 2 cubed is 8, so 8 + 7 is 15. So x = 15 when t = 2.

Now for y, the equation is y = -3t² - 6t - 5. Let me substitute t = 2 into this. Let me compute each term step by step. First, t squared is 2 squared, which is 4. Then multiply by -3: -3 * 4 = -12. Then the next term is -6t, so that's -6 * 2 = -12. And then the last term is -5.

So adding all those together: -12 (from the first term) + (-12) (from the second term) + (-5) (from the third term). Let me add them up.

-12 -12 is -24, and then -24 -5 is -29. So y = -29 when t = 2.

Wait, let me check my calculations again to make sure I didn't make a mistake.

For x: 2³ is 8, plus 7 is 15. That seems straightforward.

For y: Let me redo that. The expression is -3t² -6t -5. So plug in t = 2:

First term: -3*(2)^2 = -3*4 = -12.

Second term: -6*(2) = -12.

Third term: -5.

Adding them up: -12 -12 -5.

Hmm, so that's (-12 -12) = -24, minus 5 is -29. Yeah, that seems correct.

Wait, is there a possibility that the original equation was written differently? Let me check again.

The problem says: (x, y) = (t³ + 7, -3t² - 6t - 5). So yes, y is -3t² -6t -5.

Alternatively, maybe I should check if the signs are correct. For example, is it -3t² minus 6t minus 5? Yes. So substituting t = 2, that's -3*(4) -6*(2) -5. Yep, that's -12 -12 -5.

So that's -29.

Therefore, the coordinates when t = 2 are (15, -29).

But let me think again. Maybe there is a trick here? Sometimes problems can have hidden steps or maybe they want something else, but the question is straightforward: find the point the curve passes through at t = 2.

So unless there's some other interpretation, substituting t = 2 into both parametric equations should give the correct point.

Alternatively, maybe I need to check if the parameterization is correct? Let me check the original problem again.

"A curve is parameterized by (x,y) = (t³ + 7, -3t² - 6t - 5). Find the point the curve passes through at t = 2."

No, it's clear. So, substituting t = 2 into x and y.

Wait, let me check if I miscalculated the y-component. Maybe I can compute the entire expression as a single calculation.

Let me compute -3t² -6t -5 for t=2:

First compute t²: 4

Multiply by -3: -12

Then compute -6t: -6*2 = -12

Then the constant term: -5

So total is -12 -12 -5. Which is (-12 -12) is -24, then -24 -5 is -29.

Alternatively, maybe I can combine terms first? Let me see. The expression is -3t² -6t -5. Maybe factor something out? Let me see.

Hmm, not sure if that helps, but if I factor out a -3, it would be -3(t² + 2t) -5. Maybe not helpful. Alternatively, perhaps completing the square? Not sure. But maybe for t = 2, it's easier to just compute each term.

Alternatively, maybe check with t = 0? For example, if t = 0, then x = 0 + 7 = 7, y = -0 -0 -5 = -5. So (7, -5) when t = 0.

If t = 1, x = 1 +7 = 8, y = -3(1) -6(1) -5 = -3 -6 -5 = -14. So (8, -14).

For t = 2, we have (15, -29). Let me check t = 3? Maybe to see if the pattern makes sense?

t = 3: x = 27 + 7 = 34. y = -3*(9) -6*3 -5 = -27 -18 -5 = -50.

So (34, -50).

But maybe that's not necessary.

Alternatively, maybe check if there is a mistake in the arithmetic.

Wait, let me compute -3*(2)^2 -6*(2) -5 again:

First term: -3*(4) = -12

Second term: -6*(2) = -12

Third term: -5

Total: -12 -12 -5.

Yes, that's -29. So unless there is a sign error in the original problem. Let me check the original problem once more.

Original problem says: (x,y) = (t³ + 7, -3t² - 6t - 5). So y is definitely -3t² -6t -5.

Therefore, substituting t = 2 gives y = -3*(4) -6*(2) -5 = -12 -12 -5 = -29.

Therefore, I think that is correct.

So the point is (15, -29).

Wait, maybe the problem is expecting a different answer? Maybe I miscalculated?

Wait, let me try calculating y again:

-3t² -6t -5 at t=2:

Compute t²: 4. Multiply by -3: -12. Then compute -6t: -6*2 = -12. Then -5. So total is -12 -12 -5.

Yes, that's -29. So that's correct.

Alternatively, maybe I can think of it as:

y = -3t² -6t -5 = -3(t² + 2t) -5.

Then, completing the square for t² + 2t: t² + 2t + 1 -1 = (t + 1)^2 - 1.

So substituting back: y = -3[(t + 1)^2 - 1] -5 = -3(t + 1)^2 + 3 -5 = -3(t + 1)^2 - 2.

So that would be another way to write y. Then at t = 2, we have:

y = -3(2 + 1)^2 -2 = -3*(9) - 2 = -27 - 2 = -29.

Same result. So that's reassuring.

So, yeah, I think that's correct.

Therefore, the point on the curve when t = 2 is (15, -29).

But just to make sure, let me check once again.

x = t³ + 7. At t = 2: 8 + 7 = 15.

y: -3*(2)^2 -6*(2) -5 = -12 -12 -5 = -29.

Yes.

Therefore, I think that's the answer.

**Final Answer**
The point the curve passes through at $ t = 2 $ is \boxed{(15, -29)}.

To determine the point on the curve that the parameterization passes through when $ t = 2 $, we are given the parametric equations:

$$
x = t^3 + 7 \quad \text{and} \quad y = -3t^2 - 6t - 5
$$

We substitute the value $ t = 2 $ into both equations to compute the corresponding $ x $ and $ y $ coordinates.

---

### Step 1: Compute the $ x $-coordinate

$$
x = t^3 + 7 = 2^3 + 7 = 8 + 7 = 15
$$

---

### Step 2: Compute the $ y $-coordinate

$$
y = -3t^2 - 6t - 5
$$

Substitute $ t = 2 $:

$$
y = -3(2^2) - 6(2) - 5 = -3(4) - 12 - 5 = -12 - 12 - 5 = -29
$$

---

### Final Result

Therefore, the point on the curve that the parameterization passes through at $ t = 2 $ is:

$$
\boxed{(15, -29)}
$$

[Assistant]
Explanation:
Okay, so I need to find the point on the curve given by the parameterization (x, y) = (t³ + 7, -3t² - 6t - 5) when t = 2. Hmm, let me think.

First, parameterized curves are basically curves where both x and y are expressed in terms of a parameter, which in this case is t. So, to find the point at t = 2, I just need to substitute t = 2 into both the x and y equations, right?

Let me write that down. For x, the equation is x = t³ + 7. So plugging in t = 2, that would be x = (2)³ + 7. Let me calculate that. 2 cubed is 8, so 8 + 7 is 15. So x = 15 when t = 2.

Now for y, the equation is y = -3t² - 6t - 5. Let me substitute t = 2 into this. Let me compute each term step by step. First, t squared is 2 squared, which is 4. Then multiply by -3: -3 * 4 = -12. Then the next term is -6t, so that's -6 * 2 = -12. And then the last term is -5.

So adding all those together: -12 (from the first term) + (-12) (from the second term) + (-5) (from the third term). Let me add them up.

-12 -12 is -24, and then -24 -5 is -29. So y = -29 when t = 2.

Wait, let me check my calculations again to make sure I didn't make a mistake.

For x: 2³ is 8, plus 7 is 15. That seems straightforward.

For y: Let me redo that. The expression is -3t² -6t -5. So plug in t = 2:

First term: -3*(2)^2 = -3*4 = -12.

Second term: -6*(2) = -12.

Third term: -5.

Adding them up: -12 -12 -5.

Hmm, so that's (-12 -12) = -24, minus 5 is -29. Yeah, that seems correct.

Wait, is there a possibility that the original equation was written differently? Let me check again.

The problem says: (x, y) = (t³ + 7, -3t² - 6t - 5). So yes, y is -3t² -6t -5.

Alternatively, maybe I should check if the signs are correct. For example, is it -3t² minus 6t minus 5? Yes. So substituting t = 2, that's -3*(4) -6*(2) -5. Yep, that's -12 -12 -5.

So that's -29.

Therefore, the coordinates when t = 2 are (15, -29).

But let me think again. Maybe there is a trick here? Sometimes problems can have hidden steps or maybe they want something else, but the question is straightforward: find the point the curve passes through at t = 2.

So unless there's some other interpretation, substituting t = 2 into both parametric equations should give the correct point.

Alternatively, maybe I need to check if the parameterization is correct? Let me check the original problem again.

"A curve is parameterized by (x,y) = (t³ + 7, -3t² - 6t - 5). Find the point the curve passes through at t = 2."

No, it's clear. So, substituting t = 2 into x and y.

Wait, let me check if I miscalculated the y-component. Maybe I can compute the entire expression as a single calculation.

Let me compute -3t² -6t -5 for t=2:

First compute t²: 4

Multiply by -3: -12

Then compute -6t: -6*2 = -12

Then the constant term: -5

So total is -12 -12 -5. Which is (-12 -12) is -24, then -24 -5 is -29.

Alternatively, maybe I can combine terms first? Let me see. The expression is -3t² -6t -5. Maybe factor something out? Let me see.

Hmm, not sure if that helps, but if I factor out a -3, it would be -3(t² + 2t) -5. Maybe not helpful. Alternatively, perhaps completing the square? Not sure. But maybe for t = 2, it's easier to just compute each term.

Alternatively, maybe check with t = 0? For example, if t = 0, then x = 0 + 7 = 7, y = -0 -0 -5 = -5. So (7, -5) when t = 0.

If t = 1, x = 1 +7 = 8, y = -3(1) -6(1) -5 = -3 -6 -5 = -14. So (8, -14).

For t = 2, we have (15, -29). Let me check t = 3? Maybe to see if the pattern makes sense?

t = 3: x = 27 + 7 = 34. y = -3*(9) -6*3 -5 = -27 -18 -5 = -50.

So (34, -50).

But maybe that's not necessary.

Alternatively, maybe check if there is a mistake in the arithmetic.

Wait, let me compute -3*(2)^2 -6*(2) -5 again:

First term: -3*(4) = -12

Second term: -6*(2) = -12

Third term: -5

Total: -12 -12 -5.

Yes, that's -29. So unless there is a sign error in the original problem. Let me check the original problem once more.

Original problem says: (x,y) = (t³ + 7, -3t² - 6t - 5). So y is definitely -3t² -6t -5.

Therefore, substituting t = 2 gives y = -3*(4) -6*(2) -5 = -12 -12 -5 = -29.

Therefore, I think that is correct.

So the point is (15, -29).

Wait, maybe the problem is expecting a different answer? Maybe I miscalculated?

Wait, let me try calculating y again:

-3t² -6t -5 at t=2:

Compute t²: 4. Multiply by -3: -12. Then compute -6t: -6*2 = -12. Then -5. So total is -12 -12 -5.

Yes, that's -29. So that's correct.

Alternatively, maybe I can think of it as:

y = -3t² -6t -5 = -3(t² + 2t) -5.

Then, completing the square for t² + 2t: t² + 2t + 1 -1 = (t + 1)^2 - 1.

So substituting back: y = -3[(t + 1)^2 - 1] -5 = -3(t + 1)^2 + 3 -5 = -3(t + 1)^2 - 2.

So that would be another way to write y. Then at t = 2, we have:

y = -3(2 + 1)^2 -2 = -3*(9) - 2 = -27 - 2 = -29.

Same result. So that's reassuring.

So, yeah, I think that's correct.

Therefore, the point on the curve when t = 2 is (15, -29).

But just to make sure, let me check once again.

x = t³ + 7. At t = 2: 8 + 7 = 15.

y: -3*(2)^2 -6*(2) -5 = -12 -12 -5 = -29.

Yes.

Therefore, I think that's the answer.

**Final Answer**
The point the curve passes through at $ t = 2 $ is \boxed{(15, -29)}.

To determine the point on the curve that the parameterization passes through when $ t = 2 $, we are given the parametric equations:

$$
x = t^3 + 7 \quad \text{and} \quad y = -3t^2 - 6t - 5
$$

We substitute the value $ t = 2 $ into both equations to compute the corresponding $ x $ and $ y $ coordinates.

---

### Step 1: Compute the $ x $-coordinate

$$
x = t^3 + 7 = 2^3 + 7 = 8 + 7 = 15
$$

---

### Step 2: Compute the $ y $-coordinate

$$
y = -3t^2 - 6t - 5
$$

Substitute $ t = 2 $:

$$
y = -3(2^2) - 6(2) - 5 = -3(4) - 12 - 5 = -12 - 12 - 5 = -29
$$

---

### Final Result

Therefore, the point on the curve that the parameterization passes through at $ t = 2 $ is:

$$
\boxed{(15, -29)}
$$

Solution: At $t = 2,$ $(x,y) = (2^3 + 7, -3 \cdot 2^2 - 6 \cdot 2 - 5) = \boxed{(15,-29)}.$