Explanation:
Okay, so I need to find the value of this expression: $\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}$ when $a = 8$. Let me take it step by step.

First, let me recall what each part of the expression means. The expression has a cube root of $a^2$ multiplied by 16, and then the whole thing is raised to the power of $1/3$. Hmm, maybe I can simplify this step by step.

Since $a = 8$, I should substitute that value into the expression first. Let me write that down:

$$
\left(16\sqrt[3]{8^2}\right)^{\frac{1}{3}}
$$

Alright, let me compute the cube root part first. The cube root of $8^2$. Let me remember that cube root of a number $x$ is the same as $x^{1/3}$. So, $\sqrt[3]{8^2}$ is the same as $8^{2/3}$.

Wait, but maybe I can compute $8^2$ first and then take the cube root? Let me see.

Calculating $8^2$: 8 squared is 64. So then, $\sqrt[3]{64}$. What's the cube root of 64? Well, 4 cubed is 64 because $4 \times 4 \times 4 = 64$. So, $\sqrt[3]{64} = 4$.

So, substituting back into the expression, we have:

$$
\left(16 \times 4\right)^{\frac{1}{3}}
$$

Calculating 16 times 4: 16 * 4 is 64. So now the expression simplifies to:

$$
64^{\frac{1}{3}}
$$

Again, the cube root of 64. Wait, I already did that earlier, right? The cube root of 64 is 4. Therefore, the value should be 4.

But let me check if I did everything correctly. Maybe I should approach it differently, using exponents instead of radicals to make sure. Let me try that.

Starting again with the original expression:

$$
\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}
$$

Expressing everything with exponents:

First, note that $\sqrt[3]{a^2} = a^{2/3}$. So substituting that in:

$$
\left(16 \times a^{2/3}\right)^{1/3}
$$

Now, since $a = 8$, substitute that:

$$
\left(16 \times 8^{2/3}\right)^{1/3}
$$

Let me compute $8^{2/3}$. Again, $8^{1/3} is 2, because 2^3 = 8. Therefore, $8^{2/3} = (8^{1/3})^2 = 2^2 = 4$. So again, that gives me:

$$
(16 \times 4)^{1/3} = 64^{1/3} = 4
$$

Same result. Okay, so maybe that's correct. Wait, but let me check if there's another way to interpret the original expression. Maybe there's some order of operations I need to consider? Let me look again.

Original expression: $\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}$. The cube root is only on $a^2$, right? So the 16 is outside the cube root, so that's correct. So the entire 16 times cube root of $a^2$ is raised to the 1/3 power. So my previous steps are correct.

Alternatively, maybe I can combine the exponents? Let me try that. Let me write the entire expression as exponents.

Starting over:

Original expression: $\left(16 \times a^{2/3}\right)^{1/3}$.

Using exponent rules, when you raise a product to a power, you can distribute the exponent:

$$
16^{1/3} \times \left(a^{2/3}\right)^{1/3}
$$

Which simplifies to:

$$
16^{1/3} \times a^{(2/3 \times 1/3)} = 16^{1/3} \times a^{2/9}
$$

Wait, but then if I substitute $a = 8$, that would be:

$$
16^{1/3} \times 8^{2/9}
$$

Hmm, this seems different from before. Wait, but earlier I combined the 16 and the cube root of 8^2 first, and then raised to the 1/3 power. But when I do this approach, I'm splitting it up. Which one is correct?

Wait, let me check. If I split the exponent:

Original expression:

$$
\left(16 \times a^{2/3}\right)^{1/3} = 16^{1/3} \times a^{(2/3)(1/3)} = 16^{1/3} \times a^{2/9}
$$

But if I substitute $a = 8$, then:

$$
16^{1/3} \times 8^{2/9}
$$

But earlier, I had combined them first and got 4. Which one is correct?

Let me compute both ways.

First method:

Combine 16 and 8^{2/3} first, which is 16 * 4 = 64, then cube root of 64 is 4.

Second method:

Compute 16^{1/3} and 8^{2/9} separately, then multiply.

Let me calculate each term.

First, 16^{1/3}. 16 is 2^4, so 16^{1/3} = (2^4)^{1/3} = 2^{4/3} ≈ 2^(1 + 1/3) = 2 * 2^{1/3} ≈ 2 * 1.26 ≈ 2.52. But exact value? Not sure.

Second, 8^{2/9}. 8 is 2^3, so 8^{2/9} = (2^3)^{2/9} = 2^{6/9} = 2^{2/3} ≈ 1.5874.

Multiplying them together: 2.52 * 1.5874 ≈ approx 4.0. So maybe the exact value is 4? Because when I approximate, it's around 4. So maybe there's an exact calculation here.

Wait, maybe I can express both terms with exponents of 2 and combine them.

So 16^{1/3} = (2^4)^{1/3} = 2^{4/3}

and 8^{2/9} = (2^3)^{2/9} = 2^{6/9} = 2^{2/3}

Therefore, multiplying them: 2^{4/3} * 2^{2/3} = 2^{(4/3 + 2/3)} = 2^{6/3} = 2^2 = 4.

Oh! So that's equal to 4. So both methods give the same result, which is 4. So that's good. So even though splitting the exponents looked different, it still comes out to the same answer. So my initial calculation was correct.

Therefore, the value of the expression is 4 when a = 8.

But let me try another approach just to be thorough. Maybe simplifying the entire expression before plugging in the value of a.

Original expression: $\left(16\sqrt[3]{a^2}\right)^{1/3}$.

Let me write 16 as 2^4, and a as 8, which is 2^3. Maybe this will help.

So substituting a = 8 = 2^3:

First, inside the brackets: 16 * cube root of (a^2) = 2^4 * cube root of ( (2^3)^2 )

Calculating inside the cube root: (2^3)^2 = 2^(3*2) = 2^6

So cube root of 2^6 is 2^(6/3) = 2^2 = 4. So then:

Inside the brackets: 2^4 * 4 = 2^4 * 2^2 = 2^(4+2) = 2^6

Therefore, the entire expression becomes (2^6)^{1/3} = 2^(6*(1/3)) = 2^2 = 4.

Same answer again. Perfect!

So regardless of the method I use, I get 4. Therefore, I can be confident that the value is indeed 4.

But let me check once more for any possible mistakes. Let me compute each part step by step again:

Original problem: (16 * cube root(a²))^(1/3), with a=8.

Compute a²: 8² = 64

Compute cube root of 64: 4

Multiply by 16: 16 * 4 = 64

Take cube root of 64: 4

Yes, that's straightforward.

Alternatively, exponents:

Original expression: [16 * a^{2/3}]^{1/3} = 16^{1/3} * a^{(2/3)*(1/3)} = 16^{1/3} * a^{2/9}

With a=8:

16^{1/3} * 8^{2/9}

Expressed as powers of 2:

16 = 2^4, so 16^{1/3} = 2^{4/3}

8 = 2^3, so 8^{2/9} = (2^3)^{2/9} = 2^{6/9} = 2^{2/3}

Multiply 2^{4/3} * 2^{2/3} = 2^{(4/3 + 2/3)} = 2^{6/3} = 2^2 = 4

Same answer again.

Therefore, I think I have verified this multiple times, and each time I get 4. So unless I made a consistent error in all my calculations, which is possible but unlikely, the answer is 4.

Another way to think about this is to simplify the entire expression algebraically before plugging in the value of a. Let's try that.

Starting with the expression:

$$
\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}
$$

Expressing the cube root as an exponent:

$$
\left(16 \cdot a^{2/3}\right)^{1/3}
$$

Distribute the exponent:

$$
16^{1/3} \cdot a^{(2/3) \cdot (1/3)} = 16^{1/3} \cdot a^{2/9}
$$

So the expression simplifies to $16^{1/3} \cdot a^{2/9}$. Now, if we substitute a = 8, we can write:

$$
16^{1/3} \cdot 8^{2/9}
$$

Which, as before, can be expressed with base 2:

16 is 2^4, so 16^{1/3} is 2^{4/3}

8 is 2^3, so 8^{2/9} is (2^3)^{2/9} = 2^{6/9} = 2^{2/3}

Multiplying those gives 2^{4/3 + 2/3} = 2^{6/3} = 2^2 = 4.

Same result.

So, given that all methods consistently lead to the answer 4, I can be confident that this is the correct value.

I think the key steps here were understanding how to handle exponents and radicals, converting between radicals and exponents, and then simplifying step by step. It's also important to remember that when you have a radical raised to a power, you can express it as an exponent and then apply the rules of exponents.

Additionally, substituting the value of a early on made it easier to compute, but sometimes it's helpful to simplify algebraically first before plugging in numbers, especially when dealing with exponents. However, in this case, both approaches worked.

Another thing to note is that when simplifying exponents, it's crucial to apply the exponents correctly. For example, when raising a product to a power, each factor is raised to that power, and when multiplying terms with the same base, you add exponents. Conversely, when you have a power raised to another power, you multiply the exponents. These rules are essential in simplifying expressions like this.

In summary, by breaking down the problem into smaller, manageable parts and verifying each step through multiple approaches, I was able to confirm that the value of the expression is indeed 4 when a is 8.

**Final Answer**
The value is \boxed{4}.

To evaluate the expression
$$
\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}
$$
when $ a = 8 $, we proceed step by step with a focus on simplification and exponent rules.

---

### Step 1: Substitute $ a = 8 $

Substituting into the expression:

$$
\left(16\sqrt[3]{8^2}\right)^{\frac{1}{3}}
$$

---

### Step 2: Simplify $ \sqrt[3]{8^2} $

Recall that:
$$
\sqrt[3]{x} = x^{1/3}
$$

So,
$$
\sqrt[3]{8^2} = (8^2)^{1/3} = 8^{2/3}
$$

Now calculate $ 8^{2/3} $:

- $ 8 = 2^3 $, so:
$$
8^{2/3} = (2^3)^{2/3} = 2^{2} = 4
$$

---

### Step 3: Multiply by 16

$$
16 \cdot 4 = 64
$$

So now the expression reduces to:
$$
64^{1/3}
$$

---

### Step 4: Simplify $ 64^{1/3} $

Again, $ 64 = 2^6 $, so:
$$
64^{1/3} = (2^6)^{1/3} = 2^{6/3} = 2^2 = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of this expression: $\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}$ when $a = 8$. Let me take it step by step.

First, let me recall what each part of the expression means. The expression has a cube root of $a^2$ multiplied by 16, and then the whole thing is raised to the power of $1/3$. Hmm, maybe I can simplify this step by step.

Since $a = 8$, I should substitute that value into the expression first. Let me write that down:

$$
\left(16\sqrt[3]{8^2}\right)^{\frac{1}{3}}
$$

Alright, let me compute the cube root part first. The cube root of $8^2$. Let me remember that cube root of a number $x$ is the same as $x^{1/3}$. So, $\sqrt[3]{8^2}$ is the same as $8^{2/3}$.

Wait, but maybe I can compute $8^2$ first and then take the cube root? Let me see.

Calculating $8^2$: 8 squared is 64. So then, $\sqrt[3]{64}$. What's the cube root of 64? Well, 4 cubed is 64 because $4 \times 4 \times 4 = 64$. So, $\sqrt[3]{64} = 4$.

So, substituting back into the expression, we have:

$$
\left(16 \times 4\right)^{\frac{1}{3}}
$$

Calculating 16 times 4: 16 * 4 is 64. So now the expression simplifies to:

$$
64^{\frac{1}{3}}
$$

Again, the cube root of 64. Wait, I already did that earlier, right? The cube root of 64 is 4. Therefore, the value should be 4.

But let me check if I did everything correctly. Maybe I should approach it differently, using exponents instead of radicals to make sure. Let me try that.

Starting again with the original expression:

$$
\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}
$$

Expressing everything with exponents:

First, note that $\sqrt[3]{a^2} = a^{2/3}$. So substituting that in:

$$
\left(16 \times a^{2/3}\right)^{1/3}
$$

Now, since $a = 8$, substitute that:

$$
\left(16 \times 8^{2/3}\right)^{1/3}
$$

Let me compute $8^{2/3}$. Again, $8^{1/3} is 2, because 2^3 = 8. Therefore, $8^{2/3} = (8^{1/3})^2 = 2^2 = 4$. So again, that gives me:

$$
(16 \times 4)^{1/3} = 64^{1/3} = 4
$$

Same result. Okay, so maybe that's correct. Wait, but let me check if there's another way to interpret the original expression. Maybe there's some order of operations I need to consider? Let me look again.

Original expression: $\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}$. The cube root is only on $a^2$, right? So the 16 is outside the cube root, so that's correct. So the entire 16 times cube root of $a^2$ is raised to the 1/3 power. So my previous steps are correct.

Alternatively, maybe I can combine the exponents? Let me try that. Let me write the entire expression as exponents.

Starting over:

Original expression: $\left(16 \times a^{2/3}\right)^{1/3}$.

Using exponent rules, when you raise a product to a power, you can distribute the exponent:

$$
16^{1/3} \times \left(a^{2/3}\right)^{1/3}
$$

Which simplifies to:

$$
16^{1/3} \times a^{(2/3 \times 1/3)} = 16^{1/3} \times a^{2/9}
$$

Wait, but then if I substitute $a = 8$, that would be:

$$
16^{1/3} \times 8^{2/9}
$$

Hmm, this seems different from before. Wait, but earlier I combined the 16 and the cube root of 8^2 first, and then raised to the 1/3 power. But when I do this approach, I'm splitting it up. Which one is correct?

Wait, let me check. If I split the exponent:

Original expression:

$$
\left(16 \times a^{2/3}\right)^{1/3} = 16^{1/3} \times a^{(2/3)(1/3)} = 16^{1/3} \times a^{2/9}
$$

But if I substitute $a = 8$, then:

$$
16^{1/3} \times 8^{2/9}
$$

But earlier, I had combined them first and got 4. Which one is correct?

Let me compute both ways.

First method:

Combine 16 and 8^{2/3} first, which is 16 * 4 = 64, then cube root of 64 is 4.

Second method:

Compute 16^{1/3} and 8^{2/9} separately, then multiply.

Let me calculate each term.

First, 16^{1/3}. 16 is 2^4, so 16^{1/3} = (2^4)^{1/3} = 2^{4/3} ≈ 2^(1 + 1/3) = 2 * 2^{1/3} ≈ 2 * 1.26 ≈ 2.52. But exact value? Not sure.

Second, 8^{2/9}. 8 is 2^3, so 8^{2/9} = (2^3)^{2/9} = 2^{6/9} = 2^{2/3} ≈ 1.5874.

Multiplying them together: 2.52 * 1.5874 ≈ approx 4.0. So maybe the exact value is 4? Because when I approximate, it's around 4. So maybe there's an exact calculation here.

Wait, maybe I can express both terms with exponents of 2 and combine them.

So 16^{1/3} = (2^4)^{1/3} = 2^{4/3}

and 8^{2/9} = (2^3)^{2/9} = 2^{6/9} = 2^{2/3}

Therefore, multiplying them: 2^{4/3} * 2^{2/3} = 2^{(4/3 + 2/3)} = 2^{6/3} = 2^2 = 4.

Oh! So that's equal to 4. So both methods give the same result, which is 4. So that's good. So even though splitting the exponents looked different, it still comes out to the same answer. So my initial calculation was correct.

Therefore, the value of the expression is 4 when a = 8.

But let me try another approach just to be thorough. Maybe simplifying the entire expression before plugging in the value of a.

Original expression: $\left(16\sqrt[3]{a^2}\right)^{1/3}$.

Let me write 16 as 2^4, and a as 8, which is 2^3. Maybe this will help.

So substituting a = 8 = 2^3:

First, inside the brackets: 16 * cube root of (a^2) = 2^4 * cube root of ( (2^3)^2 )

Calculating inside the cube root: (2^3)^2 = 2^(3*2) = 2^6

So cube root of 2^6 is 2^(6/3) = 2^2 = 4. So then:

Inside the brackets: 2^4 * 4 = 2^4 * 2^2 = 2^(4+2) = 2^6

Therefore, the entire expression becomes (2^6)^{1/3} = 2^(6*(1/3)) = 2^2 = 4.

Same answer again. Perfect!

So regardless of the method I use, I get 4. Therefore, I can be confident that the value is indeed 4.

But let me check once more for any possible mistakes. Let me compute each part step by step again:

Original problem: (16 * cube root(a²))^(1/3), with a=8.

Compute a²: 8² = 64

Compute cube root of 64: 4

Multiply by 16: 16 * 4 = 64

Take cube root of 64: 4

Yes, that's straightforward.

Alternatively, exponents:

Original expression: [16 * a^{2/3}]^{1/3} = 16^{1/3} * a^{(2/3)*(1/3)} = 16^{1/3} * a^{2/9}

With a=8:

16^{1/3} * 8^{2/9}

Expressed as powers of 2:

16 = 2^4, so 16^{1/3} = 2^{4/3}

8 = 2^3, so 8^{2/9} = (2^3)^{2/9} = 2^{6/9} = 2^{2/3}

Multiply 2^{4/3} * 2^{2/3} = 2^{(4/3 + 2/3)} = 2^{6/3} = 2^2 = 4

Same answer again.

Therefore, I think I have verified this multiple times, and each time I get 4. So unless I made a consistent error in all my calculations, which is possible but unlikely, the answer is 4.

Another way to think about this is to simplify the entire expression algebraically before plugging in the value of a. Let's try that.

Starting with the expression:

$$
\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}
$$

Expressing the cube root as an exponent:

$$
\left(16 \cdot a^{2/3}\right)^{1/3}
$$

Distribute the exponent:

$$
16^{1/3} \cdot a^{(2/3) \cdot (1/3)} = 16^{1/3} \cdot a^{2/9}
$$

So the expression simplifies to $16^{1/3} \cdot a^{2/9}$. Now, if we substitute a = 8, we can write:

$$
16^{1/3} \cdot 8^{2/9}
$$

Which, as before, can be expressed with base 2:

16 is 2^4, so 16^{1/3} is 2^{4/3}

8 is 2^3, so 8^{2/9} is (2^3)^{2/9} = 2^{6/9} = 2^{2/3}

Multiplying those gives 2^{4/3 + 2/3} = 2^{6/3} = 2^2 = 4.

Same result.

So, given that all methods consistently lead to the answer 4, I can be confident that this is the correct value.

I think the key steps here were understanding how to handle exponents and radicals, converting between radicals and exponents, and then simplifying step by step. It's also important to remember that when you have a radical raised to a power, you can express it as an exponent and then apply the rules of exponents.

Additionally, substituting the value of a early on made it easier to compute, but sometimes it's helpful to simplify algebraically first before plugging in numbers, especially when dealing with exponents. However, in this case, both approaches worked.

Another thing to note is that when simplifying exponents, it's crucial to apply the exponents correctly. For example, when raising a product to a power, each factor is raised to that power, and when multiplying terms with the same base, you add exponents. Conversely, when you have a power raised to another power, you multiply the exponents. These rules are essential in simplifying expressions like this.

In summary, by breaking down the problem into smaller, manageable parts and verifying each step through multiple approaches, I was able to confirm that the value of the expression is indeed 4 when a is 8.

**Final Answer**
The value is \boxed{4}.

To evaluate the expression
$$
\left(16\sqrt[3]{a^2}\right)^{\frac{1}{3}}
$$
when $ a = 8 $, we proceed step by step with a focus on simplification and exponent rules.

---

### Step 1: Substitute $ a = 8 $

Substituting into the expression:

$$
\left(16\sqrt[3]{8^2}\right)^{\frac{1}{3}}
$$

---

### Step 2: Simplify $ \sqrt[3]{8^2} $

Recall that:
$$
\sqrt[3]{x} = x^{1/3}
$$

So,
$$
\sqrt[3]{8^2} = (8^2)^{1/3} = 8^{2/3}
$$

Now calculate $ 8^{2/3} $:

- $ 8 = 2^3 $, so:
$$
8^{2/3} = (2^3)^{2/3} = 2^{2} = 4
$$

---

### Step 3: Multiply by 16

$$
16 \cdot 4 = 64
$$

So now the expression reduces to:
$$
64^{1/3}
$$

---

### Step 4: Simplify $ 64^{1/3} $

Again, $ 64 = 2^6 $, so:
$$
64^{1/3} = (2^6)^{1/3} = 2^{6/3} = 2^2 = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

Solution: Note that $a^2 = 64$ and $\sqrt[3]{64} = 4$. Therefore, $$\left(16\sqrt[3]{a^2}\right)^{\frac {1}{3}} = \left(16 \times 4\right)^{\frac{1}{3}} = 64^\frac{1}{3} = \boxed{4}.$$


Explanation:
Okay, let me try to figure out this problem. So, Roslyn has ten boxes. Some contain pencils, some contain pens, and some contain both. The question is asking how many boxes contain neither pens nor pencils. Hmm, okay.

First, let me note down the given information. There are 10 boxes in total. Five boxes have pencils, four have pens, and two have both pens and pencils. Wait, so some boxes might be counted in both the pencil and pen groups. So, if I just add the number of boxes with pencils and pens, I might be double-counting those that have both. That makes me think of the principle of inclusion-exclusion. Yeah, right, that's the formula where the total number of elements in either set A or set B is equal to the number in A plus the number in B minus the number in both.

So, applying that here, the number of boxes that have either pencils or pens would be the number with pencils plus the number with pens minus the number with both. Let me write that down:

Number of boxes with pencils or pens = (Number with pencils) + (Number with pens) - (Number with both)

Plugging in the numbers: 5 + 4 - 2. Let me compute that. 5 plus 4 is 9, minus 2 is 7. So, there are 7 boxes that contain either pencils or pens. Therefore, the remaining boxes must contain neither. Since there are 10 boxes total, subtracting the 7 that have pencils or pens gives 10 - 7 = 3. So, 3 boxes have neither pens nor pencils.

Wait, let me check if that makes sense. Let me visualize it with a Venn diagram. There are two overlapping circles: one for pencils and one for pens. The overlap is the boxes with both. The pencils circle has 5 boxes total, but 2 of those are also in the pens circle, so the pencils-only boxes would be 5 - 2 = 3. Similarly, the pens circle has 4 boxes total, with 2 overlapping, so pens-only boxes are 4 - 2 = 2. Then the overlap is 2. So adding those up: pencils-only (3) + pens-only (2) + both (2) = 3 + 2 + 2 = 7. Yep, that's 7 boxes with pencils or pens. So total boxes are 10, so 10 - 7 = 3 boxes with neither. That seems correct.

But let me think again to make sure I didn't make a mistake. Sometimes with inclusion-exclusion, it's easy to mix up the numbers.

Total boxes = 10.

Boxes with pencils: 5. Boxes with pens: 4. Boxes with both: 2. So boxes with pencils only: 5 - 2 = 3. Boxes with pens only: 4 - 2 = 2. Then boxes with both: 2. So total boxes accounted for: 3 + 2 + 2 = 7. Then the rest, which is 10 - 7 = 3, are the ones with neither. That seems to check out.

Alternatively, another way to think about it is using the formula:

Total = (Pencils only) + (Pens only) + (Both) + (Neither)

So, substituting the known values:

10 = (5 - 2) + (4 - 2) + 2 + Neither

Which simplifies to 10 = 3 + 2 + 2 + Neither

So, 10 = 7 + Neither => Neither = 3. Yep, same result.

Alternatively, if I didn't remember the inclusion-exclusion principle, I might try to think of it as:

If there are 5 boxes with pencils, and 4 with pens, but 2 overlap, then the total unique boxes with either is 5 + 4 - 2 = 7, which again gives the same answer. So, 10 - 7 is 3.

Is there any chance that the question is trying to trick me? Like, maybe the boxes that contain both pens and pencils are being counted in both the five and the four? But the problem says "five of the boxes contain pencils, four of the boxes contain pens, and two of the boxes contain both pens and pencils." So, that two is already included in both the five and the four. So, when they say five contain pencils, that includes the two that also have pens. Similarly, the four that contain pens includes the two that also have pencils. Therefore, when you add 5 and 4, you have 9, but you've double-counted the two boxes that have both. Therefore, subtracting them once gives the correct count of unique boxes with pencils or pens. Therefore, 5 + 4 - 2 = 7, so 10 - 7 = 3.

Another way: Let me think about actual numbers. Suppose we have 10 boxes. Let's label them A, B, C, D, E, F, G, H, I, J.

Suppose boxes A, B, C, D, E have pencils. So that's 5 boxes. Now, four boxes have pens. Two of those boxes are also pencils. So, let's say boxes A, B have both pencils and pens. Then the pens are in boxes A, B, and two more boxes. Let's say boxes F, G. So pens are in A, B, F, G. So that's four boxes. Then the boxes with pencils are A, B, C, D, E. So, the ones with pencils only are C, D, E. The ones with pens only are F, G. The ones with both are A, B. Then the remaining boxes are H, I, J. So three boxes. These would be the ones with neither. So that checks out.

So in this case, there are three boxes with neither. So, the answer is 3.

But let me check if there's another way someone might approach this problem and maybe get confused. For example, if someone thought that the two boxes with both pens and pencils are separate from the five pencils and four pens. But that would be incorrect because the problem says two boxes contain both, so they are part of both groups.

Alternatively, if someone added 5 + 4 + 2, thinking that the two boxes are additional, which would be wrong. But that would lead to 11 boxes, which is more than the total of 10, so that would be impossible. So that approach is invalid. But the correct approach is subtracting the overlap.

Another possible mistake: Maybe someone subtracts the overlap twice? For example, thinking that since two boxes are both, subtract 2 from 5 and 4? But no, inclusion-exclusion is just adding and then subtracting the overlap once.

So, yeah, I think my reasoning is solid here. The answer is 3.

**Final Answer**
The number of boxes that contain neither pens nor pencils is \boxed{3}.

To determine how many boxes contain **neither pens nor pencils**, we begin by analyzing the total number of boxes and how many are occupied by either pens, pencils, or both.

---

### Step 1: Understand the Given Information

- **Total boxes** = 10
- **Boxes with pencils** = 5
- **Boxes with pens** = 4
- **Boxes with both pens and pencils** = 2

Note: These 2 boxes that have both pens and pencils are **included** in both the "pencils" and "pens" groups. Therefore, simply adding the number of pencil boxes and pen boxes would **double-count** those 2 overlapping boxes.

---

### Step 2: Apply the Inclusion-Exclusion Principle

We use the principle to find the **number of boxes that contain either pencils or pens** (or both):

$$
\text{Boxes with pencils or pens} = \text{Boxes with pencils} + \text{Boxes with pens} - \text{Boxes with both}
$$

$$
= 5 + 4 - 2 = 7
$$

So, **7 boxes** contain **either pencils, pens, or both**.

---

### Step 3: Determine Boxes with Neither Pens Nor Pencils

We now subtract the number of boxes that contain **at least one** of the items (pencils or pens) from the total number of boxes:

$$
\text{Boxes with neither} = \text{Total boxes} - \text{Boxes with pencils or pens}
$$

$$
= 10 - 7 = 3
$$

---

### Final Answer

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, Roslyn has ten boxes. Some contain pencils, some contain pens, and some contain both. The question is asking how many boxes contain neither pens nor pencils. Hmm, okay.

First, let me note down the given information. There are 10 boxes in total. Five boxes have pencils, four have pens, and two have both pens and pencils. Wait, so some boxes might be counted in both the pencil and pen groups. So, if I just add the number of boxes with pencils and pens, I might be double-counting those that have both. That makes me think of the principle of inclusion-exclusion. Yeah, right, that's the formula where the total number of elements in either set A or set B is equal to the number in A plus the number in B minus the number in both.

So, applying that here, the number of boxes that have either pencils or pens would be the number with pencils plus the number with pens minus the number with both. Let me write that down:

Number of boxes with pencils or pens = (Number with pencils) + (Number with pens) - (Number with both)

Plugging in the numbers: 5 + 4 - 2. Let me compute that. 5 plus 4 is 9, minus 2 is 7. So, there are 7 boxes that contain either pencils or pens. Therefore, the remaining boxes must contain neither. Since there are 10 boxes total, subtracting the 7 that have pencils or pens gives 10 - 7 = 3. So, 3 boxes have neither pens nor pencils.

Wait, let me check if that makes sense. Let me visualize it with a Venn diagram. There are two overlapping circles: one for pencils and one for pens. The overlap is the boxes with both. The pencils circle has 5 boxes total, but 2 of those are also in the pens circle, so the pencils-only boxes would be 5 - 2 = 3. Similarly, the pens circle has 4 boxes total, with 2 overlapping, so pens-only boxes are 4 - 2 = 2. Then the overlap is 2. So adding those up: pencils-only (3) + pens-only (2) + both (2) = 3 + 2 + 2 = 7. Yep, that's 7 boxes with pencils or pens. So total boxes are 10, so 10 - 7 = 3 boxes with neither. That seems correct.

But let me think again to make sure I didn't make a mistake. Sometimes with inclusion-exclusion, it's easy to mix up the numbers.

Total boxes = 10.

Boxes with pencils: 5. Boxes with pens: 4. Boxes with both: 2. So boxes with pencils only: 5 - 2 = 3. Boxes with pens only: 4 - 2 = 2. Then boxes with both: 2. So total boxes accounted for: 3 + 2 + 2 = 7. Then the rest, which is 10 - 7 = 3, are the ones with neither. That seems to check out.

Alternatively, another way to think about it is using the formula:

Total = (Pencils only) + (Pens only) + (Both) + (Neither)

So, substituting the known values:

10 = (5 - 2) + (4 - 2) + 2 + Neither

Which simplifies to 10 = 3 + 2 + 2 + Neither

So, 10 = 7 + Neither => Neither = 3. Yep, same result.

Alternatively, if I didn't remember the inclusion-exclusion principle, I might try to think of it as:

If there are 5 boxes with pencils, and 4 with pens, but 2 overlap, then the total unique boxes with either is 5 + 4 - 2 = 7, which again gives the same answer. So, 10 - 7 is 3.

Is there any chance that the question is trying to trick me? Like, maybe the boxes that contain both pens and pencils are being counted in both the five and the four? But the problem says "five of the boxes contain pencils, four of the boxes contain pens, and two of the boxes contain both pens and pencils." So, that two is already included in both the five and the four. So, when they say five contain pencils, that includes the two that also have pens. Similarly, the four that contain pens includes the two that also have pencils. Therefore, when you add 5 and 4, you have 9, but you've double-counted the two boxes that have both. Therefore, subtracting them once gives the correct count of unique boxes with pencils or pens. Therefore, 5 + 4 - 2 = 7, so 10 - 7 = 3.

Another way: Let me think about actual numbers. Suppose we have 10 boxes. Let's label them A, B, C, D, E, F, G, H, I, J.

Suppose boxes A, B, C, D, E have pencils. So that's 5 boxes. Now, four boxes have pens. Two of those boxes are also pencils. So, let's say boxes A, B have both pencils and pens. Then the pens are in boxes A, B, and two more boxes. Let's say boxes F, G. So pens are in A, B, F, G. So that's four boxes. Then the boxes with pencils are A, B, C, D, E. So, the ones with pencils only are C, D, E. The ones with pens only are F, G. The ones with both are A, B. Then the remaining boxes are H, I, J. So three boxes. These would be the ones with neither. So that checks out.

So in this case, there are three boxes with neither. So, the answer is 3.

But let me check if there's another way someone might approach this problem and maybe get confused. For example, if someone thought that the two boxes with both pens and pencils are separate from the five pencils and four pens. But that would be incorrect because the problem says two boxes contain both, so they are part of both groups.

Alternatively, if someone added 5 + 4 + 2, thinking that the two boxes are additional, which would be wrong. But that would lead to 11 boxes, which is more than the total of 10, so that would be impossible. So that approach is invalid. But the correct approach is subtracting the overlap.

Another possible mistake: Maybe someone subtracts the overlap twice? For example, thinking that since two boxes are both, subtract 2 from 5 and 4? But no, inclusion-exclusion is just adding and then subtracting the overlap once.

So, yeah, I think my reasoning is solid here. The answer is 3.

**Final Answer**
The number of boxes that contain neither pens nor pencils is \boxed{3}.

To determine how many boxes contain **neither pens nor pencils**, we begin by analyzing the total number of boxes and how many are occupied by either pens, pencils, or both.

---

### Step 1: Understand the Given Information

- **Total boxes** = 10
- **Boxes with pencils** = 5
- **Boxes with pens** = 4
- **Boxes with both pens and pencils** = 2

Note: These 2 boxes that have both pens and pencils are **included** in both the "pencils" and "pens" groups. Therefore, simply adding the number of pencil boxes and pen boxes would **double-count** those 2 overlapping boxes.

---

### Step 2: Apply the Inclusion-Exclusion Principle

We use the principle to find the **number of boxes that contain either pencils or pens** (or both):

$$
\text{Boxes with pencils or pens} = \text{Boxes with pencils} + \text{Boxes with pens} - \text{Boxes with both}
$$

$$
= 5 + 4 - 2 = 7
$$

So, **7 boxes** contain **either pencils, pens, or both**.

---

### Step 3: Determine Boxes with Neither Pens Nor Pencils

We now subtract the number of boxes that contain **at least one** of the items (pencils or pens) from the total number of boxes:

$$
\text{Boxes with neither} = \text{Total boxes} - \text{Boxes with pencils or pens}
$$

$$
= 10 - 7 = 3
$$

---

### Final Answer

$$
\boxed{3}
$$

Solution:
Of the 5 boxes with pencils, 2 have pens also, so $5-2=3$ have pencils only.  Similarly, $4-2 =2$ of the boxes have pens only:

[asy]
unitsize(0.05cm);
label("Pencils", (2,74));
label("Pens", (80,74));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
label("$2$", (44, 45));
label(scale(0.8)*"$3$",(28,58));
label(scale(0.8)*"$2$",(63,58));
[/asy]

That gives us $3+2+2=7$ boxes with pens, pencils, or both.  This leaves $10-7 = \boxed{3}$ with neither.


Explanation:
Okay, so I need to figure out the remainder when 2004 is divided by 12. Let me think about how to approach this. I remember that when you divide a number by another number, the remainder is what's left over after dividing as much as possible. So, for example, if I divide 10 by 3, the quotient is 3 and the remainder is 1 because 3 times 3 is 9, and 10 minus 9 is 1.

But how do I apply this to 2004 divided by 12? Well, maybe I can start by trying to divide 2004 by 12 and see what happens. Let me recall that 12 times 167 is... Hmm, wait, I need to figure out how many times 12 fits into 2004. Alternatively, maybe there's a quicker way than doing long division?

I remember something about divisibility rules. For example, to check if a number is divisible by 12, it needs to be divisible by both 3 and 4. But I'm not sure if that helps me find the remainder. Maybe I can use that to simplify the problem? Let me see.

First, let's check if 2004 is divisible by 12. If it is, then the remainder would be zero. But if not, then I need to find the remainder. Let me try dividing 2004 by 12 step by step.

Alternatively, maybe I can break down 2004 into parts that are easier to divide by 12. For instance, 2004 divided by 12 is the same as (2000 + 4) divided by 12. So that would be 2000/12 + 4/12. Let me compute each part separately.

First, 2000 divided by 12. Let me think. 12 times 166 is... Well, 12 times 160 is 1920. Then 12 times 6 is 72, so 1920 + 72 = 1992. So 12 times 166 is 1992. Then 2000 - 1992 is 8. So 2000 divided by 12 is 166 with a remainder of 8. Then, adding the 4 from the original problem, so 8 + 4 = 12. Wait, so 2000 divided by 12 has a remainder of 8, and then adding 4 gives me 12. But 12 divided by 12 is 1 with no remainder. So that would mean the total remainder is 0?

Wait, that seems conflicting. Let me check again. If I split 2004 into 2000 + 4, then 2000 divided by 12 is 166 with a remainder of 8, as above. Then 4 divided by 12 is 0 with a remainder of 4. Then, adding the remainders: 8 + 4 = 12. But since 12 is equal to 12, which is the divisor, that would mean that the total remainder is 12 - 12 = 0? Wait, no. Wait, when adding remainders, if the sum of the remainders is greater than or equal to the divisor, you have to adjust.

Let me recall that when you split a number into parts, the remainder of the whole is equal to the sum of the remainders of the parts, modulo the divisor. So in this case, the remainder of 2000/12 is 8, and the remainder of 4/12 is 4. Then 8 + 4 = 12. Then 12 divided by 12 is 1 with remainder 0. Therefore, the total remainder is 0.

So does that mean 2004 divided by 12 has a remainder of 0? Let me check with another method to confirm.

Alternatively, I can perform the division step by step. Let me try dividing 2004 by 12.

First, 12 goes into 20 once (since 12*1=12), subtract 12 from 20, you get 8. Bring down the next digit, which is 0, making it 80. 12 goes into 80 six times (12*6=72), subtract 72 from 80, you get 8. Bring down the next digit, which is 4, making it 84. 12 goes into 84 exactly 7 times (12*7=84). Subtract 84 from 84, you get 0.

So the division process gives me a quotient of 167 (since 1, then 6, then 7) and a remainder of 0. Therefore, 2004 divided by 12 is 167 with no remainder. So the remainder is 0.

Wait, but let me make sure I did the division correctly. Let me verify by multiplying 12 by 167.

Calculating 12 * 167. Let's do 12 * 160 = 1920, and 12 * 7 = 84. Adding them together, 1920 + 84 = 2004. Yes! So 12*167 is exactly 2004. Therefore, there is no remainder.

But hold on, the initial problem was to find the remainder. So according to this, the remainder is 0. But just to ensure I didn't make a mistake in my first approach.

Alternatively, maybe I can use modular arithmetic. Let me think.

What is 2004 mod 12?

I know that 12 is 3*4, so maybe I can find 2004 mod 3 and 2004 mod 4, and then use the Chinese Remainder Theorem or something? Wait, actually, since 12 is 3*4 and 3 and 4 are coprime, but maybe that's complicating.

Alternatively, I can compute 2004 divided by 12.

But maybe breaking down 2004. Let me think. 2004 divided by 12. Let me divide both numerator and denominator by 4 first.

2004 divided by 4 is 501. 12 divided by 4 is 3. Therefore, 2004 divided by 12 is equal to 501 divided by 3.

Ah, that's simpler. 501 divided by 3. Let me compute that. 3 goes into 5 once, remainder 2. Bring down 0, making it 20. 3 goes into 20 six times, remainder 2. Bring down 1, making it 21. 3 goes into 21 seven times. So the result is 167. Therefore, 501 divided by 3 is 167. Therefore, 2004 divided by 12 is 167, so the remainder is 0.

So all these methods are pointing towards the remainder being 0.

Wait, but just to be thorough, let me check with another approach. Let me use the fact that 12*167=2004.

So 12*167 = 12*(170 - 3) = 12*170 - 12*3.

Calculating 12*170: 12*100=1200, 12*70=840, so 1200+840=2040. Then 12*3=36. Therefore, 2040 - 36 = 2004. Yes! That checks out.

So 12*167=2004. Therefore, there is no remainder.

Alternatively, maybe I can use the modulo operation step by step. Let's think about 2004 divided by 12.

I know that 12*100=1200. So 2004-1200=804. Then, 12*60=720. So 804-720=84. Then, 12*7=84. Therefore, total is 100+60+7=167, with no remainder.

Another way: 12*167. Let me think of 167 as 160 + 7. 12*160=1920, 12*7=84, so 1920 + 84 = 2004.

Therefore, all methods confirm that 2004 divided by 12 is exactly 167, with no remainder.

But just to make sure, let me try dividing 2004 by 12 in another way. Let's consider that 12 times 100 is 1200. So 2004 minus 1200 is 804. Then, 12 times 60 is 720. Subtract that from 804, you get 84. 12 times 7 is 84. So total is 100 + 60 + 7 = 167. So again, no remainder.

Alternatively, let me think about the divisibility by 12. A number is divisible by 12 if it's divisible by both 3 and 4.

First, check divisibility by 3: sum the digits of 2004. 2 + 0 + 0 + 4 = 6. 6 is divisible by 3, so 2004 is divisible by 3.

Now check divisibility by 4: the last two digits of the number are 04. 04 divided by 4 is 1, which is an integer. Therefore, 2004 is divisible by 4.

Since it's divisible by both 3 and 4, it's divisible by 12. Therefore, the remainder is 0.

Yes, that's another confirmation.

So all these different methods—long division, breaking down the number, modular arithmetic, checking divisibility rules—lead me to the conclusion that 2004 divided by 12 has a remainder of 0.

Therefore, the answer should be 0.

But just to make sure I haven't overlooked something, let me try a different approach. Let me compute 2004 divided by 12 step by step using the division algorithm.

The division algorithm states that for any integers a and b (with b > 0), there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Here, a = 2004, b = 12.

So we need to find q and r such that 2004 = 12q + r, with 0 ≤ r < 12.

As I've computed before, q = 167 and r = 0.

Let me verify this equation: 12*167 + 0 = 2004.

As calculated before, 12*167=2004, so this holds.

Therefore, the remainder r is indeed 0.

Another way: Let me use the concept of modulo.

If I can find 2004 mod 12, that will give me the remainder.

I know that 2004 divided by 12 is 167, so 2004 = 12*167 + 0. Therefore, 2004 mod 12 is 0.

Alternatively, maybe I can find 2004 mod 12 by breaking down the number. Let's consider that 2004 = 2000 + 4.

As I did before, 2000 mod 12. Let me compute that.

Since 12*166 = 1992, as established earlier, then 2000 - 1992 = 8. Therefore, 2000 mod 12 = 8.

Then 4 mod 12 = 4.

Therefore, 2004 mod 12 = (8 + 4) mod 12 = 12 mod 12 = 0.

Hence, again, the remainder is 0.

So, all these different approaches confirm that the remainder is 0.

I think I have thoroughly checked this problem using multiple methods: direct division, breaking down the number, using divisibility rules, modular arithmetic, and even verifying through the division algorithm. Each time, the result is consistent.

Therefore, I can confidently conclude that the remainder when 2004 is divided by 12 is 0.

**Final Answer**
The remainder of 2004 divided by 12 is \boxed{0}.

To find the remainder of 2004 divided by 12, we can apply the concept of division with remainder, which states that for any integers $ a $ and $ b $, with $ b > 0 $, we can write:
$$
a = bq + r
$$
where $ q $ is the quotient (an integer), and $ r $ is the remainder, with $ 0 \leq r < b $.

---

### Step 1: Understanding the Problem

We are given:
- Dividend (a) = 2004
- Divisor (b) = 12

We are to find the remainder $ r $ such that:
$$
2004 = 12q + r,\quad \text{and}\quad 0 \leq r < 12
$$

---

### Step 2: Perform the Division

Let's divide 2004 by 12 step by step:

- 12 × 167 = 2004

This computation tells us that:

- The quotient $ q = 167 $
- The remainder $ r = 0 $

Because the result of the multiplication is exactly equal to the original number, the remainder is zero.

---

### Step 3: Verification via Modular Arithmetic

We can also confirm this using modular arithmetic:

$$
2004 \mod 12
$$

To compute this, we can use the following method:

- Break 2004 into 2000 + 4
- Compute: $ 2000 \mod 12 = 8 $ (since 12 × 166 = 1992, and 2000 − 1992 = 8)
- Compute: $ 4 \mod 12 = 4 $

Then:
$$
2004 \mod 12 = (8 + 4) \mod 12 = 12 \mod 12 = 0
$$

This confirms our earlier result.

---

### Step 4: Alternate Approach via Divisibility Rules

We can also check divisibility by 12 by using the fact that a number is divisible by 12 if and only if it is divisible by both 3 and 4.

- **Divisibility by 3**: Sum the digits of 2004 → 2 + 0 + 0 + 4 = 6, which is divisible by 3.
- **Divisibility by 4**: The last two digits are 04 → 4 is divisible by 4.

Thus, 2004 is divisible by 12. Hence, the remainder is 0.

---

### Final Conclusion

All methods—direct division, modular arithmetic, and divisibility rules—confirm that when 2004 is divided by 12, the remainder is:

$$
\boxed{0}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the remainder when 2004 is divided by 12. Let me think about how to approach this. I remember that when you divide a number by another number, the remainder is what's left over after dividing as much as possible. So, for example, if I divide 10 by 3, the quotient is 3 and the remainder is 1 because 3 times 3 is 9, and 10 minus 9 is 1.

But how do I apply this to 2004 divided by 12? Well, maybe I can start by trying to divide 2004 by 12 and see what happens. Let me recall that 12 times 167 is... Hmm, wait, I need to figure out how many times 12 fits into 2004. Alternatively, maybe there's a quicker way than doing long division?

I remember something about divisibility rules. For example, to check if a number is divisible by 12, it needs to be divisible by both 3 and 4. But I'm not sure if that helps me find the remainder. Maybe I can use that to simplify the problem? Let me see.

First, let's check if 2004 is divisible by 12. If it is, then the remainder would be zero. But if not, then I need to find the remainder. Let me try dividing 2004 by 12 step by step.

Alternatively, maybe I can break down 2004 into parts that are easier to divide by 12. For instance, 2004 divided by 12 is the same as (2000 + 4) divided by 12. So that would be 2000/12 + 4/12. Let me compute each part separately.

First, 2000 divided by 12. Let me think. 12 times 166 is... Well, 12 times 160 is 1920. Then 12 times 6 is 72, so 1920 + 72 = 1992. So 12 times 166 is 1992. Then 2000 - 1992 is 8. So 2000 divided by 12 is 166 with a remainder of 8. Then, adding the 4 from the original problem, so 8 + 4 = 12. Wait, so 2000 divided by 12 has a remainder of 8, and then adding 4 gives me 12. But 12 divided by 12 is 1 with no remainder. So that would mean the total remainder is 0?

Wait, that seems conflicting. Let me check again. If I split 2004 into 2000 + 4, then 2000 divided by 12 is 166 with a remainder of 8, as above. Then 4 divided by 12 is 0 with a remainder of 4. Then, adding the remainders: 8 + 4 = 12. But since 12 is equal to 12, which is the divisor, that would mean that the total remainder is 12 - 12 = 0? Wait, no. Wait, when adding remainders, if the sum of the remainders is greater than or equal to the divisor, you have to adjust.

Let me recall that when you split a number into parts, the remainder of the whole is equal to the sum of the remainders of the parts, modulo the divisor. So in this case, the remainder of 2000/12 is 8, and the remainder of 4/12 is 4. Then 8 + 4 = 12. Then 12 divided by 12 is 1 with remainder 0. Therefore, the total remainder is 0.

So does that mean 2004 divided by 12 has a remainder of 0? Let me check with another method to confirm.

Alternatively, I can perform the division step by step. Let me try dividing 2004 by 12.

First, 12 goes into 20 once (since 12*1=12), subtract 12 from 20, you get 8. Bring down the next digit, which is 0, making it 80. 12 goes into 80 six times (12*6=72), subtract 72 from 80, you get 8. Bring down the next digit, which is 4, making it 84. 12 goes into 84 exactly 7 times (12*7=84). Subtract 84 from 84, you get 0.

So the division process gives me a quotient of 167 (since 1, then 6, then 7) and a remainder of 0. Therefore, 2004 divided by 12 is 167 with no remainder. So the remainder is 0.

Wait, but let me make sure I did the division correctly. Let me verify by multiplying 12 by 167.

Calculating 12 * 167. Let's do 12 * 160 = 1920, and 12 * 7 = 84. Adding them together, 1920 + 84 = 2004. Yes! So 12*167 is exactly 2004. Therefore, there is no remainder.

But hold on, the initial problem was to find the remainder. So according to this, the remainder is 0. But just to ensure I didn't make a mistake in my first approach.

Alternatively, maybe I can use modular arithmetic. Let me think.

What is 2004 mod 12?

I know that 12 is 3*4, so maybe I can find 2004 mod 3 and 2004 mod 4, and then use the Chinese Remainder Theorem or something? Wait, actually, since 12 is 3*4 and 3 and 4 are coprime, but maybe that's complicating.

Alternatively, I can compute 2004 divided by 12.

But maybe breaking down 2004. Let me think. 2004 divided by 12. Let me divide both numerator and denominator by 4 first.

2004 divided by 4 is 501. 12 divided by 4 is 3. Therefore, 2004 divided by 12 is equal to 501 divided by 3.

Ah, that's simpler. 501 divided by 3. Let me compute that. 3 goes into 5 once, remainder 2. Bring down 0, making it 20. 3 goes into 20 six times, remainder 2. Bring down 1, making it 21. 3 goes into 21 seven times. So the result is 167. Therefore, 501 divided by 3 is 167. Therefore, 2004 divided by 12 is 167, so the remainder is 0.

So all these methods are pointing towards the remainder being 0.

Wait, but just to be thorough, let me check with another approach. Let me use the fact that 12*167=2004.

So 12*167 = 12*(170 - 3) = 12*170 - 12*3.

Calculating 12*170: 12*100=1200, 12*70=840, so 1200+840=2040. Then 12*3=36. Therefore, 2040 - 36 = 2004. Yes! That checks out.

So 12*167=2004. Therefore, there is no remainder.

Alternatively, maybe I can use the modulo operation step by step. Let's think about 2004 divided by 12.

I know that 12*100=1200. So 2004-1200=804. Then, 12*60=720. So 804-720=84. Then, 12*7=84. Therefore, total is 100+60+7=167, with no remainder.

Another way: 12*167. Let me think of 167 as 160 + 7. 12*160=1920, 12*7=84, so 1920 + 84 = 2004.

Therefore, all methods confirm that 2004 divided by 12 is exactly 167, with no remainder.

But just to make sure, let me try dividing 2004 by 12 in another way. Let's consider that 12 times 100 is 1200. So 2004 minus 1200 is 804. Then, 12 times 60 is 720. Subtract that from 804, you get 84. 12 times 7 is 84. So total is 100 + 60 + 7 = 167. So again, no remainder.

Alternatively, let me think about the divisibility by 12. A number is divisible by 12 if it's divisible by both 3 and 4.

First, check divisibility by 3: sum the digits of 2004. 2 + 0 + 0 + 4 = 6. 6 is divisible by 3, so 2004 is divisible by 3.

Now check divisibility by 4: the last two digits of the number are 04. 04 divided by 4 is 1, which is an integer. Therefore, 2004 is divisible by 4.

Since it's divisible by both 3 and 4, it's divisible by 12. Therefore, the remainder is 0.

Yes, that's another confirmation.

So all these different methods—long division, breaking down the number, modular arithmetic, checking divisibility rules—lead me to the conclusion that 2004 divided by 12 has a remainder of 0.

Therefore, the answer should be 0.

But just to make sure I haven't overlooked something, let me try a different approach. Let me compute 2004 divided by 12 step by step using the division algorithm.

The division algorithm states that for any integers a and b (with b > 0), there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. Here, a = 2004, b = 12.

So we need to find q and r such that 2004 = 12q + r, with 0 ≤ r < 12.

As I've computed before, q = 167 and r = 0.

Let me verify this equation: 12*167 + 0 = 2004.

As calculated before, 12*167=2004, so this holds.

Therefore, the remainder r is indeed 0.

Another way: Let me use the concept of modulo.

If I can find 2004 mod 12, that will give me the remainder.

I know that 2004 divided by 12 is 167, so 2004 = 12*167 + 0. Therefore, 2004 mod 12 is 0.

Alternatively, maybe I can find 2004 mod 12 by breaking down the number. Let's consider that 2004 = 2000 + 4.

As I did before, 2000 mod 12. Let me compute that.

Since 12*166 = 1992, as established earlier, then 2000 - 1992 = 8. Therefore, 2000 mod 12 = 8.

Then 4 mod 12 = 4.

Therefore, 2004 mod 12 = (8 + 4) mod 12 = 12 mod 12 = 0.

Hence, again, the remainder is 0.

So, all these different approaches confirm that the remainder is 0.

I think I have thoroughly checked this problem using multiple methods: direct division, breaking down the number, using divisibility rules, modular arithmetic, and even verifying through the division algorithm. Each time, the result is consistent.

Therefore, I can confidently conclude that the remainder when 2004 is divided by 12 is 0.

**Final Answer**
The remainder of 2004 divided by 12 is \boxed{0}.

To find the remainder of 2004 divided by 12, we can apply the concept of division with remainder, which states that for any integers $ a $ and $ b $, with $ b > 0 $, we can write:
$$
a = bq + r
$$
where $ q $ is the quotient (an integer), and $ r $ is the remainder, with $ 0 \leq r < b $.

---

### Step 1: Understanding the Problem

We are given:
- Dividend (a) = 2004
- Divisor (b) = 12

We are to find the remainder $ r $ such that:
$$
2004 = 12q + r,\quad \text{and}\quad 0 \leq r < 12
$$

---

### Step 2: Perform the Division

Let's divide 2004 by 12 step by step:

- 12 × 167 = 2004

This computation tells us that:

- The quotient $ q = 167 $
- The remainder $ r = 0 $

Because the result of the multiplication is exactly equal to the original number, the remainder is zero.

---

### Step 3: Verification via Modular Arithmetic

We can also confirm this using modular arithmetic:

$$
2004 \mod 12
$$

To compute this, we can use the following method:

- Break 2004 into 2000 + 4
- Compute: $ 2000 \mod 12 = 8 $ (since 12 × 166 = 1992, and 2000 − 1992 = 8)
- Compute: $ 4 \mod 12 = 4 $

Then:
$$
2004 \mod 12 = (8 + 4) \mod 12 = 12 \mod 12 = 0
$$

This confirms our earlier result.

---

### Step 4: Alternate Approach via Divisibility Rules

We can also check divisibility by 12 by using the fact that a number is divisible by 12 if and only if it is divisible by both 3 and 4.

- **Divisibility by 3**: Sum the digits of 2004 → 2 + 0 + 0 + 4 = 6, which is divisible by 3.
- **Divisibility by 4**: The last two digits are 04 → 4 is divisible by 4.

Thus, 2004 is divisible by 12. Hence, the remainder is 0.

---

### Final Conclusion

All methods—direct division, modular arithmetic, and divisibility rules—confirm that when 2004 is divided by 12, the remainder is:

$$
\boxed{0}
$$

Solution: An integer is divisible by 12 if and only if it is divisible by both 3 and 4.  Because $2+0+0+4=6$ is divisible by 3, 2004 is divisible by 3.  Also, the last two digits of 2004 form a multiple of 4, so 2004 is divisible by 4 as well. Therefore, 2004 is divisible by 12 and hence leaves a remainder of $\boxed{0}$ when divided by 12.


Explanation:
Okay, so there's this problem about thirty-five students attending a math club meeting. The number of girls is a multiple of 13, and there are more girls than boys. We need to find out how many boys are there. Alright, let me think step by step.

First, let me note down the given information:

Total number of students = 35

Number of girls is a multiple of 13.

Number of girls > Number of boys.

We need to find the number of boys.

Let me denote the number of girls as G and the number of boys as B. So, we have:

G + B = 35

Also, G is a multiple of 13. So, G can be 13, 26, 39, etc. But since the total number of students is 35, G can't be more than 35. So possible values for G are 13 and 26. Because 13*2=26, and 13*3=39 which is more than 35. So only two possible values for G: 13 or 26.

Now, the other condition is that there are more girls than boys. So G > B. Since G + B = 35, we can write B = 35 - G. Therefore, the condition becomes G > 35 - G. Let me solve that inequality.

G > 35 - G

Adding G to both sides:

2G > 35

Dividing by 2:

G > 17.5

So, since G must be an integer (number of people can't be a fraction), G must be at least 18. Therefore, G must be greater than 17.5, so minimum value of G is 18. However, G has to be a multiple of 13. Let me check the possible multiples of 13 that are greater than 17.5. The multiples of 13 are 13, 26, 39,... So, 13 is less than 17.5, so that's out. The next one is 26, which is greater than 17.5. So G must be 26. Therefore, the number of girls is 26. Then the number of boys is 35 - 26 = 9.

Wait, let me check if that satisfies all the conditions. Let's verify:

Total students: 26 girls + 9 boys = 35. Correct.

Number of girls is 26, which is a multiple of 13 (13*2). Correct.

Number of girls (26) is more than number of boys (9). 26 > 9, which is true.

So seems like the answer is 9 boys.

But wait, hold on. Let me check if there are any other possibilities. The multiples of 13 less than 35 are 13 and 26. We saw that 13 is less than 17.5, so G=13 would mean B=35 -13 =22. Then check if girls are more than boys: 13 > 22? No, that's false. Therefore, G cannot be 13. Hence, only possible value is 26. Therefore, boys are 9.

Therefore, the answer is 9.

But let me just make sure there isn't any other multiple of 13 that I missed. For example, if we consider 0? Well, 0 is a multiple of every number, but if there are 0 girls, then number of boys would be 35, but then girls are not more than boys. So 0 is invalid. Similarly, negative numbers? No, number of people can't be negative. So only possible multiples are 13 and 26, and 26 is the only one that satisfies the condition.

Therefore, the number of boys is 9.

Wait, but let me check once more. Is 26 the only possible multiple of 13 in the range? Let me think. 13*1=13; 13*2=26; 13*3=39. 39 is more than 35, so not possible. Therefore, yes, only 13 and 26.

Therefore, conclusion is boys are 9.

But just to make sure, let me think if there's another way someone might approach this problem and perhaps get confused. For example, someone might think that the number of girls is a multiple of 13, but maybe not necessarily an integer multiple? But no, the number of girls has to be an integer. So multiples of 13 in integers.

Another possible mistake could be miscalculating the inequality. Let me check again:

G > B

Since B = 35 - G, then:

G > 35 - G

Adding G to both sides:

2G > 35

Therefore,

G > 17.5

So G must be at least 18. Since G is a multiple of 13, the next multiple after 13 is 26, which is 13*2. So that's correct.

Alternatively, if someone thought that G must be a multiple of 13, so possible values are 13, 26, etc., but didn't consider the inequality, they might think both 13 and 26 are possible. But since with 13 girls, the number of boys is 22, which is more than girls. So the condition is violated. Therefore, only 26 girls is acceptable.

Therefore, the answer is 9 boys.

I think that's solid. No other possibilities. Therefore, I can be confident that the answer is 9.

**Final Answer**
The number of boys at the meeting is \boxed{9}.

To solve the problem, we are given the following:

- A total of **35 students** attend the meeting.
- The **number of girls** is a **multiple of 13**.
- There are **more girls than boys** at the meeting.

---

### Step 1: Define the Variables

Let:

- $ G $ = number of girls
- $ B $ = number of boys

We know:

$$
G + B = 35
$$
$$
G > B
$$
$$
G \text{ is a multiple of } 13
$$

We can rewrite $ B $ in terms of $ G $:

$$
B = 35 - G
$$

Using the inequality $ G > B $, substitute $ B $:

$$
G > 35 - G
$$
$$
2G > 35
$$
$$
G > 17.5
$$

Since $ G $ must be an integer, the smallest possible value for $ G $ satisfying this is **18**.

---

### Step 2: Find Valid Multiples of 13 Greater Than 17.5

We are told that $ G $ is a **multiple of 13**. The multiples of 13 are:

- $ 13 \times 1 = 13 $
- $ 13 \times 2 = 26 $
- $ 13 \times 3 = 39 $ (too large, since total students is 35)

So the possible values for $ G $ are:

- **13**
- **26**

Now, we check which of these satisfy the condition $ G > 17.5 $:

- $ G = 13 $ → $ G < 17.5 $ → **Does not satisfy**
- $ G = 26 $ → $ G > 17.5 $ → **Satisfies**

So the only valid value for $ G $ is **26**.

---

### Step 3: Compute the Number of Boys

$$
B = 35 - G = 35 - 26 = 9
$$

Now, we verify all conditions:

- Total students: $ 26 + 9 = 35 $ ✅
- $ G = 26 $ is a multiple of 13 ✅
- $ G = 26 > 9 = B $, so more girls than boys ✅

---

### Final Answer

$$
\boxed{9}
$$

[Assistant]
Explanation:
Okay, so there's this problem about thirty-five students attending a math club meeting. The number of girls is a multiple of 13, and there are more girls than boys. We need to find out how many boys are there. Alright, let me think step by step.

First, let me note down the given information:

Total number of students = 35

Number of girls is a multiple of 13.

Number of girls > Number of boys.

We need to find the number of boys.

Let me denote the number of girls as G and the number of boys as B. So, we have:

G + B = 35

Also, G is a multiple of 13. So, G can be 13, 26, 39, etc. But since the total number of students is 35, G can't be more than 35. So possible values for G are 13 and 26. Because 13*2=26, and 13*3=39 which is more than 35. So only two possible values for G: 13 or 26.

Now, the other condition is that there are more girls than boys. So G > B. Since G + B = 35, we can write B = 35 - G. Therefore, the condition becomes G > 35 - G. Let me solve that inequality.

G > 35 - G

Adding G to both sides:

2G > 35

Dividing by 2:

G > 17.5

So, since G must be an integer (number of people can't be a fraction), G must be at least 18. Therefore, G must be greater than 17.5, so minimum value of G is 18. However, G has to be a multiple of 13. Let me check the possible multiples of 13 that are greater than 17.5. The multiples of 13 are 13, 26, 39,... So, 13 is less than 17.5, so that's out. The next one is 26, which is greater than 17.5. So G must be 26. Therefore, the number of girls is 26. Then the number of boys is 35 - 26 = 9.

Wait, let me check if that satisfies all the conditions. Let's verify:

Total students: 26 girls + 9 boys = 35. Correct.

Number of girls is 26, which is a multiple of 13 (13*2). Correct.

Number of girls (26) is more than number of boys (9). 26 > 9, which is true.

So seems like the answer is 9 boys.

But wait, hold on. Let me check if there are any other possibilities. The multiples of 13 less than 35 are 13 and 26. We saw that 13 is less than 17.5, so G=13 would mean B=35 -13 =22. Then check if girls are more than boys: 13 > 22? No, that's false. Therefore, G cannot be 13. Hence, only possible value is 26. Therefore, boys are 9.

Therefore, the answer is 9.

But let me just make sure there isn't any other multiple of 13 that I missed. For example, if we consider 0? Well, 0 is a multiple of every number, but if there are 0 girls, then number of boys would be 35, but then girls are not more than boys. So 0 is invalid. Similarly, negative numbers? No, number of people can't be negative. So only possible multiples are 13 and 26, and 26 is the only one that satisfies the condition.

Therefore, the number of boys is 9.

Wait, but let me check once more. Is 26 the only possible multiple of 13 in the range? Let me think. 13*1=13; 13*2=26; 13*3=39. 39 is more than 35, so not possible. Therefore, yes, only 13 and 26.

Therefore, conclusion is boys are 9.

But just to make sure, let me think if there's another way someone might approach this problem and perhaps get confused. For example, someone might think that the number of girls is a multiple of 13, but maybe not necessarily an integer multiple? But no, the number of girls has to be an integer. So multiples of 13 in integers.

Another possible mistake could be miscalculating the inequality. Let me check again:

G > B

Since B = 35 - G, then:

G > 35 - G

Adding G to both sides:

2G > 35

Therefore,

G > 17.5

So G must be at least 18. Since G is a multiple of 13, the next multiple after 13 is 26, which is 13*2. So that's correct.

Alternatively, if someone thought that G must be a multiple of 13, so possible values are 13, 26, etc., but didn't consider the inequality, they might think both 13 and 26 are possible. But since with 13 girls, the number of boys is 22, which is more than girls. So the condition is violated. Therefore, only 26 girls is acceptable.

Therefore, the answer is 9 boys.

I think that's solid. No other possibilities. Therefore, I can be confident that the answer is 9.

**Final Answer**
The number of boys at the meeting is \boxed{9}.

To solve the problem, we are given the following:

- A total of **35 students** attend the meeting.
- The **number of girls** is a **multiple of 13**.
- There are **more girls than boys** at the meeting.

---

### Step 1: Define the Variables

Let:

- $ G $ = number of girls
- $ B $ = number of boys

We know:

$$
G + B = 35
$$
$$
G > B
$$
$$
G \text{ is a multiple of } 13
$$

We can rewrite $ B $ in terms of $ G $:

$$
B = 35 - G
$$

Using the inequality $ G > B $, substitute $ B $:

$$
G > 35 - G
$$
$$
2G > 35
$$
$$
G > 17.5
$$

Since $ G $ must be an integer, the smallest possible value for $ G $ satisfying this is **18**.

---

### Step 2: Find Valid Multiples of 13 Greater Than 17.5

We are told that $ G $ is a **multiple of 13**. The multiples of 13 are:

- $ 13 \times 1 = 13 $
- $ 13 \times 2 = 26 $
- $ 13 \times 3 = 39 $ (too large, since total students is 35)

So the possible values for $ G $ are:

- **13**
- **26**

Now, we check which of these satisfy the condition $ G > 17.5 $:

- $ G = 13 $ → $ G < 17.5 $ → **Does not satisfy**
- $ G = 26 $ → $ G > 17.5 $ → **Satisfies**

So the only valid value for $ G $ is **26**.

---

### Step 3: Compute the Number of Boys

$$
B = 35 - G = 35 - 26 = 9
$$

Now, we verify all conditions:

- Total students: $ 26 + 9 = 35 $ ✅
- $ G = 26 $ is a multiple of 13 ✅
- $ G = 26 > 9 = B $, so more girls than boys ✅

---

### Final Answer

$$
\boxed{9}
$$

Solution: We assume the number of girls is a nonnegative integer and no greater than 35. The nonnegative multiples of 13 less than 35 are 0, 13 and 26. As the number of girls is greater than the number of boys, the only valid choice is for there to be 26 girls. That leaves $35-26 = \boxed{9}$ boys at the meeting.


Explanation:
Okay, so I need to figure out the value of 5 star 1, where the star operation is defined as a star b equals 9a plus 2b minus ab plus 5. Let me write that down to make sure I understand the problem correctly.

The operation is given by:
$$ a \star b = 9a + 2b - ab + 5 $$

And I need to compute:
$$ 5 \star 1 $$

Alright, so let me substitute a = 5 and b = 1 into the formula. Let me do that step by step to avoid mistakes.

First, let me rewrite the formula with a and b:
$$ a \star b = 9a + 2b - ab + 5 $$

So substituting a = 5 and b = 1:
$$ 5 \star 1 = 9(5) + 2(1) - (5)(1) + 5 $$

Now let me compute each term one by one.

Starting with the first term: 9 multiplied by 5. Let me calculate 9*5. Well, 9*5 is 45. Okay, so that term is 45.

Next term is 2 multiplied by 1. That's straightforward: 2*1 = 2. So the second term is 2.

Third term is - (5*1). Let me compute 5*1 first, which is 5, and then apply the negative sign. So that becomes -5.

The last term is just +5. So that's a positive 5.

Now let me add all these terms together:

First term: 45

Second term: +2 → 45 + 2 = 47

Third term: -5 → 47 - 5 = 42

Fourth term: +5 → 42 + 5 = 47

Wait, so adding them up step by step gives me 47? Let me verify that again to make sure I didn't make any arithmetic errors.

Starting over:

9*5 = 45

2*1 = 2

5*1 = 5, and since there's a negative sign, it's -5.

Then the last term is +5.

So putting it all together: 45 + 2 is 47, minus 5 is 42, plus 5 is 47. Yeah, that seems right.

But let me check if I substituted correctly. The original formula is 9a + 2b - ab + 5. So substituting a=5 and b=1:

9*5 + 2*1 - (5*1) + 5. Yes, that's exactly what I did. So that's 45 + 2 - 5 + 5. Wait, hold on, is it 45 + 2 - 5 + 5? Let me check that again.

Wait, 9a is 45, 2b is 2, -ab is -(5*1) which is -5, and then +5. So yes, it's 45 + 2 - 5 + 5. So combining the constants:

45 + 2 is 47, then -5 + 5 is 0. Therefore, 47 + 0 = 47. So that's 47. Therefore, the answer should be 47.

Wait, but let me check once again. Maybe I should rearrange the terms. Let me write all the terms:

9a = 45

2b = 2

-ab = -5

+5 = +5

So total is 45 + 2 - 5 + 5. Let me group the -5 and +5. They cancel each other out, right? -5 + 5 = 0. Therefore, total is 45 + 2 + 0 = 47. Yep, same result.

Alternatively, maybe I can think of the formula as 9a + 2b - ab + 5. Maybe there's a way to factor it or simplify it, but since the problem just asks for 5 star 1, perhaps there's no need. But just for my own understanding, maybe I can try to rearrange the formula.

Looking at the formula: 9a + 2b - ab + 5. Let me see if I can factor anything here. Let's see:

Looking at the terms with a: 9a - ab. That can be factored as a(9 - b). Then the remaining terms are 2b + 5. So overall:

a(9 - b) + 2b + 5.

So if I substitute a = 5 and b = 1, that would be:

5*(9 - 1) + 2*1 + 5 = 5*8 + 2 + 5 = 40 + 2 + 5 = 47. Same result. So that confirms it again.

Alternatively, maybe I can think of the formula in another way. Let me check:

Original formula: 9a + 2b - ab + 5. Maybe rearranged as:

(9a - ab) + (2b + 5). Which is the same as before. So again, same calculation.

Alternatively, maybe I can write it as:

Let me think if there's another way. For example, combining terms differently. But I don't think it's necessary here because substitution seems straightforward.

But just to be thorough, let me check once again with substitution:

Original definition:

a ⋆ b = 9a + 2b - ab + 5

So for a=5 and b=1:

Compute each term:

9a: 9*5 = 45

2b: 2*1 = 2

-ab: -(5*1) = -5

+5: +5

Add them all: 45 + 2 - 5 + 5. Let me compute step by step:

Start with 45.

Add 2: 45 + 2 = 47.

Subtract 5: 47 - 5 = 42.

Add 5: 42 + 5 = 47.

Yep, same result again. So I think 47 is the correct answer here.

Wait, but let me check if there is any possibility that I misread the original operation. Let me confirm the problem statement again.

"If a ⋆ b = 9a + 2b - ab + 5, what is the value of 5 ⋆ 1?"

Yes, that's correct. So the operation is defined as 9a + 2b minus ab plus 5. So no, I didn't misread it. So substituting a=5 and b=1 gives 45 + 2 - 5 + 5 = 47.

Alternatively, perhaps I can check with another approach. Let me think: Maybe there's a different interpretation of the formula? For example, is the formula 9a + 2b - ab + 5 or is there a different order of operations?

Let me check the original problem again: "a ⋆ b = 9a + 2b - ab + 5". The way it's written, it's 9a plus 2b minus ab plus 5. So the operations are all addition and subtraction, so order of operations would be multiplication before addition and subtraction. So, no, there's no ambiguity here. So the calculation is correct.

Alternatively, maybe I can think of the formula as:

a ⋆ b = 9a + 2b - ab + 5 = (9a - ab) + (2b + 5) = a(9 - b) + (2b + 5). Which is the same as before.

Alternatively, maybe I can plug into a different order. For example, maybe I can compute 9a + 2b first, then subtract ab, then add 5. Let me try that:

For a=5, b=1:

Compute 9a + 2b: 45 + 2 = 47.

Then subtract ab: 47 - 5 = 42.

Then add 5: 42 + 5 = 47. Same result. Yep, that's the same as before.

So regardless of how I compute it, I get 47. Therefore, I think the answer is 47.

Is there any chance I could have messed up the signs? Let me check again:

Original formula: 9a + 2b - ab + 5. So the terms are:

+9a, +2b, -ab, +5. So substituting:

+9*5, +2*1, -5*1, +5. Which is +45 +2 -5 +5. So yes, that's correct. So the negatives and positives are accounted for.

Alternatively, maybe if I had misread the formula as 9a + 2b - (ab +5) instead of 9a + 2b - ab + 5? But no, the original problem says "-ab +5", so it's minus ab plus 5. So that would be different if it was - (ab + 5), but that's not the case here.

So I think my answer is correct. Therefore, the value of 5 ⋆ 1 is 47.

**Final Answer**
The value of $5 \star 1$ is \boxed{47}.

To determine the value of $ 5 \star 1 $, we begin with the definition of the custom operation $ \star $:

$$
a \star b = 9a + 2b - ab + 5
$$

---

### Step 1: Substitution of values

We are asked to compute:

$$
5 \star 1
$$

This means substituting $ a = 5 $ and $ b = 1 $ into the given formula:

$$
5 \star 1 = 9(5) + 2(1) - (5)(1) + 5
$$

---

### Step 2: Compute each term

- $ 9(5) = 45 $
- $ 2(1) = 2 $
- $ 5(1) = 5 $, and since this term is subtracted, it becomes $-5$
- The constant $ +5 $ remains as is

So, we have:

$$
45 + 2 - 5 + 5
$$

---

### Step 3: Simplify the expression

First, combine the constants step by step:

- $ 45 + 2 = 47 $
- $ 47 - 5 = 42 $
- $ 42 + 5 = 47 $

---

### Final Answer

Thus, the value of $ 5 \star 1 $ is:

$$
\boxed{47}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the value of 5 star 1, where the star operation is defined as a star b equals 9a plus 2b minus ab plus 5. Let me write that down to make sure I understand the problem correctly.

The operation is given by:
$$ a \star b = 9a + 2b - ab + 5 $$

And I need to compute:
$$ 5 \star 1 $$

Alright, so let me substitute a = 5 and b = 1 into the formula. Let me do that step by step to avoid mistakes.

First, let me rewrite the formula with a and b:
$$ a \star b = 9a + 2b - ab + 5 $$

So substituting a = 5 and b = 1:
$$ 5 \star 1 = 9(5) + 2(1) - (5)(1) + 5 $$

Now let me compute each term one by one.

Starting with the first term: 9 multiplied by 5. Let me calculate 9*5. Well, 9*5 is 45. Okay, so that term is 45.

Next term is 2 multiplied by 1. That's straightforward: 2*1 = 2. So the second term is 2.

Third term is - (5*1). Let me compute 5*1 first, which is 5, and then apply the negative sign. So that becomes -5.

The last term is just +5. So that's a positive 5.

Now let me add all these terms together:

First term: 45

Second term: +2 → 45 + 2 = 47

Third term: -5 → 47 - 5 = 42

Fourth term: +5 → 42 + 5 = 47

Wait, so adding them up step by step gives me 47? Let me verify that again to make sure I didn't make any arithmetic errors.

Starting over:

9*5 = 45

2*1 = 2

5*1 = 5, and since there's a negative sign, it's -5.

Then the last term is +5.

So putting it all together: 45 + 2 is 47, minus 5 is 42, plus 5 is 47. Yeah, that seems right.

But let me check if I substituted correctly. The original formula is 9a + 2b - ab + 5. So substituting a=5 and b=1:

9*5 + 2*1 - (5*1) + 5. Yes, that's exactly what I did. So that's 45 + 2 - 5 + 5. Wait, hold on, is it 45 + 2 - 5 + 5? Let me check that again.

Wait, 9a is 45, 2b is 2, -ab is -(5*1) which is -5, and then +5. So yes, it's 45 + 2 - 5 + 5. So combining the constants:

45 + 2 is 47, then -5 + 5 is 0. Therefore, 47 + 0 = 47. So that's 47. Therefore, the answer should be 47.

Wait, but let me check once again. Maybe I should rearrange the terms. Let me write all the terms:

9a = 45

2b = 2

-ab = -5

+5 = +5

So total is 45 + 2 - 5 + 5. Let me group the -5 and +5. They cancel each other out, right? -5 + 5 = 0. Therefore, total is 45 + 2 + 0 = 47. Yep, same result.

Alternatively, maybe I can think of the formula as 9a + 2b - ab + 5. Maybe there's a way to factor it or simplify it, but since the problem just asks for 5 star 1, perhaps there's no need. But just for my own understanding, maybe I can try to rearrange the formula.

Looking at the formula: 9a + 2b - ab + 5. Let me see if I can factor anything here. Let's see:

Looking at the terms with a: 9a - ab. That can be factored as a(9 - b). Then the remaining terms are 2b + 5. So overall:

a(9 - b) + 2b + 5.

So if I substitute a = 5 and b = 1, that would be:

5*(9 - 1) + 2*1 + 5 = 5*8 + 2 + 5 = 40 + 2 + 5 = 47. Same result. So that confirms it again.

Alternatively, maybe I can think of the formula in another way. Let me check:

Original formula: 9a + 2b - ab + 5. Maybe rearranged as:

(9a - ab) + (2b + 5). Which is the same as before. So again, same calculation.

Alternatively, maybe I can write it as:

Let me think if there's another way. For example, combining terms differently. But I don't think it's necessary here because substitution seems straightforward.

But just to be thorough, let me check once again with substitution:

Original definition:

a ⋆ b = 9a + 2b - ab + 5

So for a=5 and b=1:

Compute each term:

9a: 9*5 = 45

2b: 2*1 = 2

-ab: -(5*1) = -5

+5: +5

Add them all: 45 + 2 - 5 + 5. Let me compute step by step:

Start with 45.

Add 2: 45 + 2 = 47.

Subtract 5: 47 - 5 = 42.

Add 5: 42 + 5 = 47.

Yep, same result again. So I think 47 is the correct answer here.

Wait, but let me check if there is any possibility that I misread the original operation. Let me confirm the problem statement again.

"If a ⋆ b = 9a + 2b - ab + 5, what is the value of 5 ⋆ 1?"

Yes, that's correct. So the operation is defined as 9a + 2b minus ab plus 5. So no, I didn't misread it. So substituting a=5 and b=1 gives 45 + 2 - 5 + 5 = 47.

Alternatively, perhaps I can check with another approach. Let me think: Maybe there's a different interpretation of the formula? For example, is the formula 9a + 2b - ab + 5 or is there a different order of operations?

Let me check the original problem again: "a ⋆ b = 9a + 2b - ab + 5". The way it's written, it's 9a plus 2b minus ab plus 5. So the operations are all addition and subtraction, so order of operations would be multiplication before addition and subtraction. So, no, there's no ambiguity here. So the calculation is correct.

Alternatively, maybe I can think of the formula as:

a ⋆ b = 9a + 2b - ab + 5 = (9a - ab) + (2b + 5) = a(9 - b) + (2b + 5). Which is the same as before.

Alternatively, maybe I can plug into a different order. For example, maybe I can compute 9a + 2b first, then subtract ab, then add 5. Let me try that:

For a=5, b=1:

Compute 9a + 2b: 45 + 2 = 47.

Then subtract ab: 47 - 5 = 42.

Then add 5: 42 + 5 = 47. Same result. Yep, that's the same as before.

So regardless of how I compute it, I get 47. Therefore, I think the answer is 47.

Is there any chance I could have messed up the signs? Let me check again:

Original formula: 9a + 2b - ab + 5. So the terms are:

+9a, +2b, -ab, +5. So substituting:

+9*5, +2*1, -5*1, +5. Which is +45 +2 -5 +5. So yes, that's correct. So the negatives and positives are accounted for.

Alternatively, maybe if I had misread the formula as 9a + 2b - (ab +5) instead of 9a + 2b - ab + 5? But no, the original problem says "-ab +5", so it's minus ab plus 5. So that would be different if it was - (ab + 5), but that's not the case here.

So I think my answer is correct. Therefore, the value of 5 ⋆ 1 is 47.

**Final Answer**
The value of $5 \star 1$ is \boxed{47}.

To determine the value of $ 5 \star 1 $, we begin with the definition of the custom operation $ \star $:

$$
a \star b = 9a + 2b - ab + 5
$$

---

### Step 1: Substitution of values

We are asked to compute:

$$
5 \star 1
$$

This means substituting $ a = 5 $ and $ b = 1 $ into the given formula:

$$
5 \star 1 = 9(5) + 2(1) - (5)(1) + 5
$$

---

### Step 2: Compute each term

- $ 9(5) = 45 $
- $ 2(1) = 2 $
- $ 5(1) = 5 $, and since this term is subtracted, it becomes $-5$
- The constant $ +5 $ remains as is

So, we have:

$$
45 + 2 - 5 + 5
$$

---

### Step 3: Simplify the expression

First, combine the constants step by step:

- $ 45 + 2 = 47 $
- $ 47 - 5 = 42 $
- $ 42 + 5 = 47 $

---

### Final Answer

Thus, the value of $ 5 \star 1 $ is:

$$
\boxed{47}
$$

Solution: From the defined function, we know that $5\star 1 = 9(5)+2(1)-(5)(1)+5= 45+2-5+5=\boxed{47}$.