Explanation:
Okay, so I need to find the total surface area of a cube that has a volume of 1 cubic foot. Hmm, let me think. First, I remember that the volume of a cube is calculated by cubing the length of one of its sides. So, if the volume is 1 cubic foot, then the length of each side must be the cube root of 1. Let me write that down.

Volume of a cube = side³. So, if Volume = 1 ft³, then side = ∛1. The cube root of 1 is 1, right? So each side of the cube is 1 foot long. Wait, but the question asks for the surface area in square inches. Oh, right! I need to convert feet to inches because the answer needs to be in square inches.

Alright, so first, let me confirm the side length. If the volume is 1 cubic foot, then the side length is 1 foot. Now, converting feet to inches. Since 1 foot is 12 inches, the side length is 12 inches. Got it.

Now, the surface area of a cube. I remember that a cube has 6 faces, each face being a square. The area of one face is side length squared. So, total surface area is 6 times the area of one face. Let me write that formula down.

Total Surface Area = 6 × (side length)².

So, substituting the side length in inches, which is 12 inches. Let me calculate that.

First, calculate the area of one face: 12 inches × 12 inches = 144 square inches. Then multiply by 6 for all six faces. So, 6 × 144. Let me do that multiplication.

Hmm, 6 × 144. Well, 144 × 6. Let me break it down. 100 × 6 = 600, 40 × 6 = 240, 4 × 6 = 24. Adding those together: 600 + 240 = 840, plus 24 is 864. So, 864 square inches.

Wait, let me check that again. Maybe there's a simpler way. 144 × 6. Since 144 × 5 = 720, and then add 144 more, so 720 + 144 = 864. Yep, same result. So the total surface area is 864 square inches.

But let me just make sure I didn't skip any steps or make any conversion errors. Let me go through it again.

Original volume is 1 cubic foot. Convert that to cubic inches? Wait, maybe another way to approach it is to convert the volume to cubic inches first, then find the side length in inches, then compute the surface area. Let me try that method to cross-verify.

1 cubic foot is equal to how many cubic inches? Since 1 foot is 12 inches, then 1 cubic foot is 12 inches × 12 inches × 12 inches. Let me calculate that. 12 × 12 is 144, and 144 × 12. Let me compute that. 144 × 10 = 1440, 144 × 2 = 288, so total is 1440 + 288 = 1728. So, 1 cubic foot is 1728 cubic inches.

So, the volume is 1728 cubic inches. Then, the side length is the cube root of 1728. Let me calculate that. Hmm, 12³ is 1728, right? Because 12 × 12 = 144, 144 × 12 = 1728. Yes, so the side length is 12 inches. Which matches what I had before. So, then the surface area is 6 × (12²) = 6 × 144 = 864 square inches.

Okay, so both methods give me the same answer. That seems consistent. So, I think that's correct.

But just to make sure, let me recall the formulas again. Volume of a cube is s³, surface area is 6s². So, if I have volume in cubic feet, converting to cubic inches is necessary because the answer needs to be in square inches.

Alternatively, if I had kept the side length in feet, then calculated the surface area in square feet, and then converted that to square inches. Let me try that approach as well.

Original side length is 1 foot. Surface area is 6 × (1 foot)² = 6 square feet. Then, converting square feet to square inches. Since 1 foot is 12 inches, 1 square foot is 12 inches × 12 inches = 144 square inches. Therefore, 6 square feet is 6 × 144 = 864 square inches.

Same answer again. So, regardless of the method, I get 864 square inches. Therefore, I think that's the correct answer.

I don't see any mistakes in my reasoning. All three methods—converting volume to cubic inches first, converting side length to inches first, and converting surface area from square feet to square inches—lead me to the same result. So, I feel confident that the total surface area is 864 square inches.

**Final Answer**
The total surface area of the cube is \boxed{864} square inches.

To determine the total surface area of a cube with a volume of 1 cubic foot, we proceed step by step, ensuring all units are consistent and the calculations are accurate.

---

### Step 1: Understand the Volume and Side Length

The volume $ V $ of a cube is given by the formula:

$$
V = s^3
$$

where $ s $ is the length of one side of the cube. Given that the volume is 1 cubic foot:

$$
s^3 = 1 \text{ ft}^3 \Rightarrow s = \sqrt[3]{1} = 1 \text{ ft}
$$

So, the side length of the cube is 1 foot.

---

### Step 2: Convert Side Length to Inches

Since the final answer must be in square inches, we convert the side length from feet to inches. Recall:

$$
1 \text{ ft} = 12 \text{ in}
$$

Thus, the side length in inches is:

$$
s = 12 \text{ in}
$$

---

### Step 3: Calculate the Surface Area

The surface area $ A $ of a cube is given by:

$$
A = 6s^2
$$

Substituting $ s = 12 $ inches:

$$
A = 6 \times (12)^2 = 6 \times 144 = 864 \text{ in}^2
$$

---

### Final Answer

$$
\boxed{864}
$$

The total surface area of the cube is **864 square inches**.

[Assistant]
Explanation:
Okay, so I need to find the total surface area of a cube that has a volume of 1 cubic foot. Hmm, let me think. First, I remember that the volume of a cube is calculated by cubing the length of one of its sides. So, if the volume is 1 cubic foot, then the length of each side must be the cube root of 1. Let me write that down.

Volume of a cube = side³. So, if Volume = 1 ft³, then side = ∛1. The cube root of 1 is 1, right? So each side of the cube is 1 foot long. Wait, but the question asks for the surface area in square inches. Oh, right! I need to convert feet to inches because the answer needs to be in square inches.

Alright, so first, let me confirm the side length. If the volume is 1 cubic foot, then the side length is 1 foot. Now, converting feet to inches. Since 1 foot is 12 inches, the side length is 12 inches. Got it.

Now, the surface area of a cube. I remember that a cube has 6 faces, each face being a square. The area of one face is side length squared. So, total surface area is 6 times the area of one face. Let me write that formula down.

Total Surface Area = 6 × (side length)².

So, substituting the side length in inches, which is 12 inches. Let me calculate that.

First, calculate the area of one face: 12 inches × 12 inches = 144 square inches. Then multiply by 6 for all six faces. So, 6 × 144. Let me do that multiplication.

Hmm, 6 × 144. Well, 144 × 6. Let me break it down. 100 × 6 = 600, 40 × 6 = 240, 4 × 6 = 24. Adding those together: 600 + 240 = 840, plus 24 is 864. So, 864 square inches.

Wait, let me check that again. Maybe there's a simpler way. 144 × 6. Since 144 × 5 = 720, and then add 144 more, so 720 + 144 = 864. Yep, same result. So the total surface area is 864 square inches.

But let me just make sure I didn't skip any steps or make any conversion errors. Let me go through it again.

Original volume is 1 cubic foot. Convert that to cubic inches? Wait, maybe another way to approach it is to convert the volume to cubic inches first, then find the side length in inches, then compute the surface area. Let me try that method to cross-verify.

1 cubic foot is equal to how many cubic inches? Since 1 foot is 12 inches, then 1 cubic foot is 12 inches × 12 inches × 12 inches. Let me calculate that. 12 × 12 is 144, and 144 × 12. Let me compute that. 144 × 10 = 1440, 144 × 2 = 288, so total is 1440 + 288 = 1728. So, 1 cubic foot is 1728 cubic inches.

So, the volume is 1728 cubic inches. Then, the side length is the cube root of 1728. Let me calculate that. Hmm, 12³ is 1728, right? Because 12 × 12 = 144, 144 × 12 = 1728. Yes, so the side length is 12 inches. Which matches what I had before. So, then the surface area is 6 × (12²) = 6 × 144 = 864 square inches.

Okay, so both methods give me the same answer. That seems consistent. So, I think that's correct.

But just to make sure, let me recall the formulas again. Volume of a cube is s³, surface area is 6s². So, if I have volume in cubic feet, converting to cubic inches is necessary because the answer needs to be in square inches.

Alternatively, if I had kept the side length in feet, then calculated the surface area in square feet, and then converted that to square inches. Let me try that approach as well.

Original side length is 1 foot. Surface area is 6 × (1 foot)² = 6 square feet. Then, converting square feet to square inches. Since 1 foot is 12 inches, 1 square foot is 12 inches × 12 inches = 144 square inches. Therefore, 6 square feet is 6 × 144 = 864 square inches.

Same answer again. So, regardless of the method, I get 864 square inches. Therefore, I think that's the correct answer.

I don't see any mistakes in my reasoning. All three methods—converting volume to cubic inches first, converting side length to inches first, and converting surface area from square feet to square inches—lead me to the same result. So, I feel confident that the total surface area is 864 square inches.

**Final Answer**
The total surface area of the cube is \boxed{864} square inches.

To determine the total surface area of a cube with a volume of 1 cubic foot, we proceed step by step, ensuring all units are consistent and the calculations are accurate.

---

### Step 1: Understand the Volume and Side Length

The volume $ V $ of a cube is given by the formula:

$$
V = s^3
$$

where $ s $ is the length of one side of the cube. Given that the volume is 1 cubic foot:

$$
s^3 = 1 \text{ ft}^3 \Rightarrow s = \sqrt[3]{1} = 1 \text{ ft}
$$

So, the side length of the cube is 1 foot.

---

### Step 2: Convert Side Length to Inches

Since the final answer must be in square inches, we convert the side length from feet to inches. Recall:

$$
1 \text{ ft} = 12 \text{ in}
$$

Thus, the side length in inches is:

$$
s = 12 \text{ in}
$$

---

### Step 3: Calculate the Surface Area

The surface area $ A $ of a cube is given by:

$$
A = 6s^2
$$

Substituting $ s = 12 $ inches:

$$
A = 6 \times (12)^2 = 6 \times 144 = 864 \text{ in}^2
$$

---

### Final Answer

$$
\boxed{864}
$$

The total surface area of the cube is **864 square inches**.

Solution:
Recall that  $$1 \mbox{ foot} = 12 \mbox{ inches}$$

Hence $$1 \mbox{ foot}^2 = 12^2 \mbox{ inches}^2 = 144 \mbox{ inches}^2$$

Finally, remember the formula $V = l \times w \times h$, that is, Volume is the product of length, width, and height.

The length, height, and width for a cube are equal, so the cube we are given has sides of length $1 \mbox{ feet}$. Now, a cube has 6 faces, so the surface area of our cube is $$6 \times (1 \mbox{ foot} \times 1 \mbox{ foot}) = 6 \mbox{ feet}^2$$

Now, convert: $$ 6 \mbox{ feet}^2 \frac{144 \mbox{ inches}^2}{1 \mbox{ foot}^2} = 864 \mbox{ inches}^2$$

So, our final answer is $\boxed{864 \mbox{ inches}^2}$


Explanation:
Okay, so I need to figure out how many sides a regular polygon has if its interior angle is 6.5 times the measure of its exterior angle. Hmm, let me recall what I know about polygons.

First, I remember that in any polygon, the sum of the interior and exterior angles at each vertex is 180 degrees. That's because they are supplementary angles. So, if I let the measure of an exterior angle be, say, x degrees, then the interior angle would be 180 - x degrees.

But the problem says the interior angle is 6.5 times the exterior angle. So, maybe I can set up an equation with that information. Let me write that down:

Interior angle = 6.5 × Exterior angle

But since Interior angle + Exterior angle = 180°, I can substitute the interior angle with 180 - Exterior angle. Wait, let me check that again. If the interior angle is 6.5 times the exterior angle, then:

Interior angle = 6.5 × Exterior angle

But also, Interior angle + Exterior angle = 180°, so substituting the first equation into the second:

6.5 × Exterior angle + Exterior angle = 180°

Let me combine those terms. 6.5x + x = 180°, where x is the exterior angle. That would be 7.5x = 180°, right? So then x = 180° / 7.5. Let me calculate that.

Dividing 180 by 7.5... Hmm, 7.5 times 24 is 180 because 7.5 × 20 = 150, and 7.5 × 4 = 30, so 150 + 30 = 180. So x = 24°. Therefore, each exterior angle is 24 degrees.

Now, since the polygon is regular, all exterior angles are equal, and the sum of all exterior angles of any polygon is 360 degrees. So, if each exterior angle is 24°, then the number of sides, let's call it n, is 360° divided by the measure of each exterior angle.

So n = 360° / 24°. Let me compute that. 360 divided by 24. Well, 24 × 15 = 360, right? Because 24 × 10 = 240, 24 × 5 = 120, so 240 + 120 = 360. Therefore, n = 15.

Wait, so the polygon has 15 sides? Let me check if that makes sense.

If it's a regular 15-gon, each exterior angle is 360/15 = 24°, so the interior angle would be 180 - 24 = 156°. Then, is 156° equal to 6.5 times 24°? Let me check: 24 × 6.5.

Well, 24 × 6 = 144, and 24 × 0.5 = 12, so 144 + 12 = 156. Yes, that's correct. So the interior angle is indeed 6.5 times the exterior angle.

Therefore, the number of sides is 15.

But let me just go through the steps again to make sure I didn't skip anything or make a mistake.

Starting with the relationship between interior and exterior angles. Since they are supplementary, interior + exterior = 180. Given that interior = 6.5 × exterior, so substituting gives 6.5x + x = 180, which is 7.5x = 180. Solving for x gives x = 24. Then, since the sum of exterior angles is 360, number of sides is 360 / 24 = 15.

Yes, that seems consistent.

Alternatively, maybe I can approach it using formulas for interior and exterior angles.

The formula for the measure of an interior angle of a regular polygon is [(n - 2) × 180] / n.

The measure of an exterior angle is 360 / n.

Given that the interior angle is 6.5 times the exterior angle, so:

[(n - 2)/n] × 180 = 6.5 × (360 / n)

Let me write that equation:

[(n - 2)/n] × 180 = 6.5 × (360 / n)

Multiply both sides by n to eliminate denominators:

(n - 2) × 180 = 6.5 × 360

Compute the right side: 6.5 × 360. Let me calculate that.

Well, 6 × 360 = 2160, and 0.5 × 360 = 180, so total is 2160 + 180 = 2340.

So left side is 180(n - 2) = 2340.

Divide both sides by 180:

n - 2 = 2340 / 180

Calculate that. Let me divide numerator and denominator by 10: 234 / 18.

Divide numerator and denominator by 6: 39 / 3 = 13.

Wait, 234 divided by 18: 18 × 13 = 234. Yes, so 2340 / 180 = 13.

Therefore, n - 2 = 13, so n = 15.

Same answer. So that confirms it.

Therefore, the polygon has 15 sides.

I think that's solid. Both methods lead to the same answer, so I feel confident that 15 is the correct number of sides.

**Final Answer**
The polygon has \boxed{15} sides.

To determine the number of sides in a regular polygon where the measure of an interior angle is **6.5 times** the measure of an exterior angle, we can use the fundamental relationships between interior and exterior angles in polygons.

---

### Step 1: Understand the Relationship Between Interior and Exterior Angles

In any polygon, the **interior and exterior angles at each vertex are supplementary**, meaning:

$$
\text{Interior angle} + \text{Exterior angle} = 180^\circ
$$

Let the measure of an **exterior angle** be $ x $. Then the **interior angle** is:

$$
180^\circ - x
$$

According to the problem, the interior angle is **6.5 times** the exterior angle:

$$
180 - x = 6.5x
$$

---

### Step 2: Solve for the Exterior Angle

$$
180 - x = 6.5x \\
180 = 7.5x \\
x = \frac{180}{7.5} = 24^\circ
$$

So, each **exterior angle** measures $ 24^\circ $.

---

### Step 3: Use the Sum of Exterior Angles

In any **regular polygon**, the **sum of all exterior angles** is always $ 360^\circ $, and since all exterior angles are equal in a regular polygon, the number of sides $ n $ is:

$$
n = \frac{360^\circ}{\text{measure of one exterior angle}} = \frac{360}{24} = 15
$$

---

### Step 4: Verification

Let’s verify the result:

- Each **exterior angle** = $ 24^\circ $
- Each **interior angle** = $ 180 - 24 = 156^\circ $
- Check if $ 156 = 6.5 \times 24 $:

$$
6.5 \times 24 = 156
$$

This confirms the relationship holds.

---

### Final Answer

$$
\boxed{15}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many sides a regular polygon has if its interior angle is 6.5 times the measure of its exterior angle. Hmm, let me recall what I know about polygons.

First, I remember that in any polygon, the sum of the interior and exterior angles at each vertex is 180 degrees. That's because they are supplementary angles. So, if I let the measure of an exterior angle be, say, x degrees, then the interior angle would be 180 - x degrees.

But the problem says the interior angle is 6.5 times the exterior angle. So, maybe I can set up an equation with that information. Let me write that down:

Interior angle = 6.5 × Exterior angle

But since Interior angle + Exterior angle = 180°, I can substitute the interior angle with 180 - Exterior angle. Wait, let me check that again. If the interior angle is 6.5 times the exterior angle, then:

Interior angle = 6.5 × Exterior angle

But also, Interior angle + Exterior angle = 180°, so substituting the first equation into the second:

6.5 × Exterior angle + Exterior angle = 180°

Let me combine those terms. 6.5x + x = 180°, where x is the exterior angle. That would be 7.5x = 180°, right? So then x = 180° / 7.5. Let me calculate that.

Dividing 180 by 7.5... Hmm, 7.5 times 24 is 180 because 7.5 × 20 = 150, and 7.5 × 4 = 30, so 150 + 30 = 180. So x = 24°. Therefore, each exterior angle is 24 degrees.

Now, since the polygon is regular, all exterior angles are equal, and the sum of all exterior angles of any polygon is 360 degrees. So, if each exterior angle is 24°, then the number of sides, let's call it n, is 360° divided by the measure of each exterior angle.

So n = 360° / 24°. Let me compute that. 360 divided by 24. Well, 24 × 15 = 360, right? Because 24 × 10 = 240, 24 × 5 = 120, so 240 + 120 = 360. Therefore, n = 15.

Wait, so the polygon has 15 sides? Let me check if that makes sense.

If it's a regular 15-gon, each exterior angle is 360/15 = 24°, so the interior angle would be 180 - 24 = 156°. Then, is 156° equal to 6.5 times 24°? Let me check: 24 × 6.5.

Well, 24 × 6 = 144, and 24 × 0.5 = 12, so 144 + 12 = 156. Yes, that's correct. So the interior angle is indeed 6.5 times the exterior angle.

Therefore, the number of sides is 15.

But let me just go through the steps again to make sure I didn't skip anything or make a mistake.

Starting with the relationship between interior and exterior angles. Since they are supplementary, interior + exterior = 180. Given that interior = 6.5 × exterior, so substituting gives 6.5x + x = 180, which is 7.5x = 180. Solving for x gives x = 24. Then, since the sum of exterior angles is 360, number of sides is 360 / 24 = 15.

Yes, that seems consistent.

Alternatively, maybe I can approach it using formulas for interior and exterior angles.

The formula for the measure of an interior angle of a regular polygon is [(n - 2) × 180] / n.

The measure of an exterior angle is 360 / n.

Given that the interior angle is 6.5 times the exterior angle, so:

[(n - 2)/n] × 180 = 6.5 × (360 / n)

Let me write that equation:

[(n - 2)/n] × 180 = 6.5 × (360 / n)

Multiply both sides by n to eliminate denominators:

(n - 2) × 180 = 6.5 × 360

Compute the right side: 6.5 × 360. Let me calculate that.

Well, 6 × 360 = 2160, and 0.5 × 360 = 180, so total is 2160 + 180 = 2340.

So left side is 180(n - 2) = 2340.

Divide both sides by 180:

n - 2 = 2340 / 180

Calculate that. Let me divide numerator and denominator by 10: 234 / 18.

Divide numerator and denominator by 6: 39 / 3 = 13.

Wait, 234 divided by 18: 18 × 13 = 234. Yes, so 2340 / 180 = 13.

Therefore, n - 2 = 13, so n = 15.

Same answer. So that confirms it.

Therefore, the polygon has 15 sides.

I think that's solid. Both methods lead to the same answer, so I feel confident that 15 is the correct number of sides.

**Final Answer**
The polygon has \boxed{15} sides.

To determine the number of sides in a regular polygon where the measure of an interior angle is **6.5 times** the measure of an exterior angle, we can use the fundamental relationships between interior and exterior angles in polygons.

---

### Step 1: Understand the Relationship Between Interior and Exterior Angles

In any polygon, the **interior and exterior angles at each vertex are supplementary**, meaning:

$$
\text{Interior angle} + \text{Exterior angle} = 180^\circ
$$

Let the measure of an **exterior angle** be $ x $. Then the **interior angle** is:

$$
180^\circ - x
$$

According to the problem, the interior angle is **6.5 times** the exterior angle:

$$
180 - x = 6.5x
$$

---

### Step 2: Solve for the Exterior Angle

$$
180 - x = 6.5x \\
180 = 7.5x \\
x = \frac{180}{7.5} = 24^\circ
$$

So, each **exterior angle** measures $ 24^\circ $.

---

### Step 3: Use the Sum of Exterior Angles

In any **regular polygon**, the **sum of all exterior angles** is always $ 360^\circ $, and since all exterior angles are equal in a regular polygon, the number of sides $ n $ is:

$$
n = \frac{360^\circ}{\text{measure of one exterior angle}} = \frac{360}{24} = 15
$$

---

### Step 4: Verification

Let’s verify the result:

- Each **exterior angle** = $ 24^\circ $
- Each **interior angle** = $ 180 - 24 = 156^\circ $
- Check if $ 156 = 6.5 \times 24 $:

$$
6.5 \times 24 = 156
$$

This confirms the relationship holds.

---

### Final Answer

$$
\boxed{15}
$$

Solution:
The measure of an interior angle of a regular $n$-gon is $\frac{180(n-2)}{n}$ degrees and the measure of an exterior angle is $\frac{360}{n}$ degrees.  Solving  \[
\frac{180(n-2)}{n}=6.5\cdot\left(\frac{360}{n}\right),
\] we find $n=\boxed{15}$.


Explanation:
Okay, so I need to find the inverse of 997 modulo 1000. That means I need to find an integer x such that 997 * x ≡ 1 mod 1000. Hmm, inverses modulo... I remember that for an inverse to exist, the number and the modulus must be coprime. Let me check if 997 and 1000 are coprime.

First, let me recall that two numbers are coprime if their greatest common divisor (gcd) is 1. So, I need to compute gcd(997, 1000). Let me do that.

Well, 1000 divided by 997 is 1 with a remainder of 3. Then, gcd(997, 1000) is the same as gcd(997, 3). Now, 997 divided by 3 is 332 with a remainder of 1, since 3*332 = 996, and 997 - 996 = 1. Then, gcd(3, 1). Which is 1. So yes, they are coprime. Great, so an inverse exists.

Now, how do I find the inverse? I remember that one way is to use the Extended Euclidean Algorithm. Let me try that.

The Extended Euclidean Algorithm not only finds the gcd but also finds integers x and y such that ax + by = gcd(a, b). In this case, since gcd(997, 1000) = 1, we can find x and y such that 997x + 1000y = 1. Then, x would be the inverse of 997 modulo 1000.

Let me set up the algorithm. Let me write down the steps of the Euclidean algorithm first, then backtrack to find x and y.

Starting with 1000 and 997:

1000 = 1 * 997 + 3
997 = 332 * 3 + 1
3 = 3 * 1 + 0

So the gcd is 1, as we saw. Now, let's work backwards to express 1 as a combination of 1000 and 997.

From the second equation:
1 = 997 - 332 * 3

But from the first equation, 3 = 1000 - 1 * 997. Substitute that into the above equation:
1 = 997 - 332 * (1000 - 1 * 997)
Let me expand that:
1 = 997 - 332 * 1000 + 332 * 997
Combine like terms:
1 = (1 + 332) * 997 - 332 * 1000
Which is:
1 = 333 * 997 - 332 * 1000

So, rearranged, this is:
1 = 333 * 997 + (-332) * 1000

Therefore, comparing to the equation ax + by = 1, we have x = 333 and y = -332.

Therefore, the inverse of 997 modulo 1000 is 333. Wait, but let me check that. Let me compute 997 * 333 mod 1000.

But before I do that, let me make sure I didn't make a mistake in the algebra. Let me verify the steps again.

Starting from the second equation:
1 = 997 - 332 * 3
Then, substitute 3 = 1000 - 997:
1 = 997 - 332*(1000 - 997)
= 997 - 332*1000 + 332*997
= (1 + 332)*997 - 332*1000
= 333*997 - 332*1000
Yes, that seems correct.

So, according to this, 333 is the coefficient for 997, so 333 is the inverse. But let me check with actual multiplication.

Compute 997 * 333. Let me compute that.

First, note that 997 is 1000 - 3. So, 997 * 333 = (1000 - 3) * 333 = 1000*333 - 3*333.

So, modulo 1000, the 1000*333 term is 0, so we have -3*333 mod 1000.

Compute -3*333: that is -999. Then, -999 mod 1000 is 1, since -999 + 1000 = 1.

Yes! So 997 * 333 ≡ 1 mod 1000. Therefore, 333 is indeed the inverse.

Wait, but let me check with another method just to be sure. Maybe using another approach?

Alternatively, since we are working modulo 1000, maybe we can find x such that 997x ≡ 1 mod 1000.

But 997 ≡ -3 mod 1000. Therefore, the equation becomes -3x ≡ 1 mod 1000. Which is equivalent to 3x ≡ -1 mod 1000, or 3x ≡ 999 mod 1000.

So solving 3x ≡ 999 mod 1000.

Divide both sides by gcd(3, 1000). Since gcd(3, 1000) is 1, we can find the inverse of 3 mod 1000 and multiply.

So, inverse of 3 mod 1000. Let me find that.

Find y such that 3y ≡ 1 mod 1000.

Again, using Extended Euclidean Algorithm.

Let me do that.

Find gcd(3, 1000):

1000 = 3 * 333 + 1
3 = 1 * 3 + 0

So gcd is 1. Then backtracking:

1 = 1000 - 3 * 333

Therefore, 1 = (-333)*3 + 1*1000. Therefore, inverse of 3 mod 1000 is -333 mod 1000.

-333 mod 1000 is 1000 - 333 = 667.

Therefore, inverse of 3 is 667.

Therefore, going back to 3x ≡ 999 mod 1000. Multiply both sides by 667:

x ≡ 999 * 667 mod 1000.

Compute 999 * 667 mod 1000.

Again, note that 999 ≡ -1 mod 1000. Therefore, this is (-1)*667 mod 1000 = -667 mod 1000 = 1000 - 667 = 333.

So again, x ≡ 333 mod 1000. Therefore, same result.

Therefore, 333 is indeed the inverse.

Alternatively, maybe I can check with another approach. Let me try to compute 997 * x mod 1000 = 1.

But since 997 ≡ -3 mod 1000, so (-3)x ≡ 1 mod 1000. So 3x ≡ -1 mod 1000. Which is same as 3x ≡ 999 mod 1000.

Alternatively, maybe I can solve 3x ≡ 999 mod 1000.

Divide both sides by 3? Wait, but division in modular arithmetic is multiplication by inverse.

But since we already found that inverse of 3 is 667, so x ≡ 999 * 667 mod 1000.

But maybe compute 999 * 667 mod 1000.

Alternatively, since 999 ≡ -1, so (-1)*667 = -667 ≡ 333 mod 1000.

Same result.

Alternatively, maybe compute 3x ≡ 999 mod 1000. Let me divide both sides by something?

But since 3 and 1000 are coprime, we can divide.

But maybe I can solve for x by trial? Although that might take longer.

Alternatively, note that 3x ≡ 999 mod 1000. Let me divide both sides by gcd(3, 1000) = 1, so the equation is equivalent to x ≡ 999 * 3^{-1} mod 1000. Which is again 333.

Alternatively, maybe think of 3x ≡ 999 mod 1000. Let me write this as 3x - 1000k = 999 for some integer k.

So 3x - 1000k = 999.

Let me rearrange: 3x = 1000k + 999.

So x = (1000k + 999)/3.

We need x to be integer, so 1000k + 999 must be divisible by 3.

Compute 1000 mod 3. 1000 divided by 3 is 333*3 + 1, so 1000 ≡ 1 mod 3. Similarly, 999 is 3*333, so 999 ≡ 0 mod 3.

Therefore, 1000k + 999 ≡ 1*k + 0 ≡ k mod 3.

We need this to be 0 mod 3, so k ≡ 0 mod 3.

Therefore, let k = 3m for some integer m.

Then x = (1000*(3m) + 999)/3 = 1000m + 333.

Therefore, x ≡ 333 mod 1000.

Therefore, the smallest positive solution is 333.

So again, same answer.

Therefore, seems like 333 is the inverse.

But just to be thorough, let me compute 997 * 333 and check if it's 1 mod 1000.

Compute 997 * 333.

Well, 997 * 333. Let me compute this.

First, note that 1000 * 333 = 333,000.

Subtract 3 * 333 = 999.

Therefore, 997 * 333 = 333,000 - 999 = 332,001.

Now, 332,001 divided by 1000 is 332 with a remainder of 1. Therefore, 332,001 mod 1000 is 1.

Yes! So 997 * 333 ≡ 1 mod 1000.

Therefore, 333 is indeed the inverse.

Alternatively, maybe I can check with another number. Let me try multiplying 997 by 333 and see.

But since I already did that, and it worked, I think that's solid.

Alternatively, maybe check with another inverse? For example, if I think of 997 ≡ -3 mod 1000, so inverse of -3 is -inverse(3). Since inverse of 3 is 667, inverse of -3 is -667 mod 1000, which is 1000 - 667 = 333. Yep, same answer.

Therefore, all methods lead to 333.

Therefore, the answer is 333.

But just to make sure, let me try another approach. Maybe using the fact that 997 is congruent to -3 mod 1000, so solving (-3)x ≡ 1 mod 1000.

Which is equivalent to 3x ≡ -1 mod 1000.

So 3x ≡ 999 mod 1000.

Let me try to solve this equation step by step.

Let me write 3x = 999 + 1000k for some integer k.

So x = (999 + 1000k)/3.

We need x to be integer.

So 999 + 1000k must be divisible by 3.

As before, 999 is divisible by 3 (since 9+9+9=27, which is divisible by 3). 1000k mod 3 is (1*k) mod 3, since 1000 ≡ 1 mod 3.

Therefore, 999 + 1000k ≡ 0 + k ≡ k mod 3.

Therefore, k must be ≡ 0 mod 3.

So let k = 3m.

Then x = (999 + 1000*(3m))/3 = 999/3 + (1000*3m)/3 = 333 + 1000m.

Therefore, x ≡ 333 mod 1000.

Same result.

Therefore, regardless of the method, the inverse is 333.

Alternatively, maybe I can use the fact that 997 is congruent to -3 mod 1000, so inverse is -1/3 mod 1000. Which is -667 mod 1000, which is 333.

So, all methods converge.

Therefore, I can be confident that the inverse of 997 modulo 1000 is 333.

But just to make sure, let me check with another value. Let me pick a number close to 333 and see.

For example, let me try 333.

Compute 997 * 333.

As before, 997 * 333 = 332,001.

Divide by 1000: 332,001 / 1000 = 332.001. So the remainder is 1.

Yes, so 332,001 mod 1000 is 1.

Alternatively, if I take 333 + 1000 = 1333. Then 997 * 1333 would be 997*(1000 + 333) = 997*1000 + 997*333. Which is 997000 + 332001 = 1,329,001. Then mod 1000, that's 1. So same result.

Therefore, 333 is the correct inverse.

Alternatively, maybe check with another number. Let me try 333 - 1000 = -667. Let me check if -667 is also an inverse.

Compute 997 * (-667) mod 1000.

Again, 997 ≡ -3, so (-3)*(-667) = 2001. 2001 mod 1000 is 1.

Yes! So -667 is also an inverse, but since we need the answer between 0 and 999, we take -667 mod 1000, which is 333.

Therefore, same answer.

Therefore, I think I can be confident that 333 is the correct inverse.

Is there any other way to check? Maybe using another modulus?

Alternatively, since 1000 = 8 * 125, and 997 is congruent to -3 mod 8 and mod 125. Maybe use Chinese Remainder Theorem?

Let me try that approach.

Find x such that 997x ≡ 1 mod 8 and 997x ≡ 1 mod 125. Then combine the solutions.

First, compute 997 mod 8.

Divide 997 by 8: 8*124 = 992, so 997 - 992 = 5. Therefore, 997 ≡ 5 mod 8.

So equation 1: 5x ≡ 1 mod 8.

Solve for x.

Multiply both sides by inverse of 5 mod 8.

What is inverse of 5 mod 8?

Find y such that 5y ≡ 1 mod 8.

Testing y=5: 5*5=25≡1 mod8. Yes. So inverse is 5.

Therefore, x ≡ 5*1 ≡ 5 mod 8.

So x ≡ 5 mod 8.

Now, equation 2: 997x ≡ 1 mod 125.

Compute 997 mod 125.

Divide 997 by 125. 125*7 = 875. 997 - 875 = 122. So 997 ≡ 122 mod 125.

So equation: 122x ≡ 1 mod 125.

Need to solve for x.

Find inverse of 122 mod 125.

Note that 122 ≡ -3 mod 125. So inverse of -3 mod 125.

Which is same as -inverse(3) mod 125.

Find inverse of 3 mod 125.

Find y such that 3y ≡ 1 mod 125.

Use Extended Euclidean Algorithm.

Compute gcd(3, 125):

125 = 41*3 + 2
3 = 1*2 + 1
2 = 2*1 + 0

So gcd is 1. Now backtracking:

1 = 3 - 1*2
But 2 = 125 - 41*3, so substitute:

1 = 3 - 1*(125 - 41*3) = 3 - 125 + 41*3 = 42*3 - 125.

Therefore, 1 = 42*3 - 125*1.

Therefore, inverse of 3 mod 125 is 42.

Therefore, inverse of -3 mod 125 is -42 mod 125.

-42 mod 125 is 125 - 42 = 83.

Therefore, inverse of 122 mod 125 is 83.

Therefore, x ≡ 83 mod 125.

So now, we have:

x ≡ 5 mod 8
x ≡ 83 mod 125

We need to find x such that these two congruences hold. Then x will be the inverse mod 1000.

Let me solve this system.

Let x = 125k + 83 for some integer k.

Substitute into first equation: 125k + 83 ≡ 5 mod 8.

Compute 125 mod 8: 125 / 8 = 15*8 + 5, so 125 ≡ 5 mod 8.

Similarly, 83 mod 8: 83 / 8 = 10*8 + 3, so 83 ≡ 3 mod 8.

Therefore, equation becomes: 5k + 3 ≡ 5 mod 8
Subtract 3: 5k ≡ 2 mod 8
Solve for k: 5k ≡ 2 mod 8.

Multiply both sides by inverse of 5 mod 8, which we know is 5.

So k ≡ 2*5 ≡ 10 ≡ 2 mod 8.

Therefore, k = 8m + 2 for some integer m.

Therefore, x = 125*(8m + 2) + 83 = 1000m + 250 + 83 = 1000m + 333.

Therefore, x ≡ 333 mod 1000.

Therefore, the solution is 333 mod 1000.

Therefore, again, same answer.

Therefore, using Chinese Remainder Theorem also gives 333.

Therefore, this confirms that 333 is indeed the inverse.

So, after multiple methods: Extended Euclidean Algorithm, solving via modular inverses, Chinese Remainder Theorem, direct computation, all lead to the same answer.

Therefore, I can be confident that the inverse of 997 modulo 1000 is 333.

**Final Answer**
The inverse of $997$ modulo $1000$ is $\boxed{333}$.

To find the modular inverse of $ 997 $ modulo $ 1000 $, we are looking for an integer $ x $ such that:

$$
997x \equiv 1 \pmod{1000}
$$

---

### Step 1: Confirm that an inverse exists

An inverse exists if and only if the number and the modulus are coprime. That is, $ \gcd(997, 1000) = 1 $.

We compute:

$$
\gcd(997, 1000) = \gcd(997, 1000 - 997) = \gcd(997, 3)
$$
$$
\gcd(997, 3) = \gcd(3, 997 \mod 3) = \gcd(3, 1) = 1
$$

So, $ 997 $ and $ 1000 $ are coprime, and an inverse exists.

---

### Step 2: Use the Extended Euclidean Algorithm

We apply the Extended Euclidean Algorithm to find integers $ x $ and $ y $ such that:

$$
997x + 1000y = 1
$$

We perform the Euclidean algorithm:

$$
\begin{align*}
1000 &= 1 \cdot 997 + 3 \\
997 &= 332 \cdot 3 + 1 \\
3 &= 3 \cdot 1 + 0
\end{align*}
$$

Now, we backtrack to express 1 as a linear combination of 997 and 1000:

$$
1 = 997 - 332 \cdot 3
$$
$$
\text{But } 3 = 1000 - 1 \cdot 997
$$
$$
\Rightarrow 1 = 997 - 332 \cdot (1000 - 997) = 997 - 332 \cdot 1000 + 332 \cdot 997
$$
$$
\Rightarrow 1 = (1 + 332) \cdot 997 - 332 \cdot 1000 = 333 \cdot 997 - 332 \cdot 1000
$$

So, the coefficient of 997 is $ x = 333 $, which is the modular inverse of 997 modulo 1000.

---

### Step 3: Verify the result

We verify that:

$$
997 \cdot 333 \equiv 1 \pmod{1000}
$$

Note that $ 997 \equiv -3 \pmod{1000} $, so:

$$
997 \cdot 333 \equiv (-3) \cdot 333 = -999 \equiv 1 \pmod{1000}
$$

This confirms that $ 333 $ is indeed the inverse.

---

### Final Answer

$$
\boxed{333}
$$

[Assistant]
Explanation:
Okay, so I need to find the inverse of 997 modulo 1000. That means I need to find an integer x such that 997 * x ≡ 1 mod 1000. Hmm, inverses modulo... I remember that for an inverse to exist, the number and the modulus must be coprime. Let me check if 997 and 1000 are coprime.

First, let me recall that two numbers are coprime if their greatest common divisor (gcd) is 1. So, I need to compute gcd(997, 1000). Let me do that.

Well, 1000 divided by 997 is 1 with a remainder of 3. Then, gcd(997, 1000) is the same as gcd(997, 3). Now, 997 divided by 3 is 332 with a remainder of 1, since 3*332 = 996, and 997 - 996 = 1. Then, gcd(3, 1). Which is 1. So yes, they are coprime. Great, so an inverse exists.

Now, how do I find the inverse? I remember that one way is to use the Extended Euclidean Algorithm. Let me try that.

The Extended Euclidean Algorithm not only finds the gcd but also finds integers x and y such that ax + by = gcd(a, b). In this case, since gcd(997, 1000) = 1, we can find x and y such that 997x + 1000y = 1. Then, x would be the inverse of 997 modulo 1000.

Let me set up the algorithm. Let me write down the steps of the Euclidean algorithm first, then backtrack to find x and y.

Starting with 1000 and 997:

1000 = 1 * 997 + 3
997 = 332 * 3 + 1
3 = 3 * 1 + 0

So the gcd is 1, as we saw. Now, let's work backwards to express 1 as a combination of 1000 and 997.

From the second equation:
1 = 997 - 332 * 3

But from the first equation, 3 = 1000 - 1 * 997. Substitute that into the above equation:
1 = 997 - 332 * (1000 - 1 * 997)
Let me expand that:
1 = 997 - 332 * 1000 + 332 * 997
Combine like terms:
1 = (1 + 332) * 997 - 332 * 1000
Which is:
1 = 333 * 997 - 332 * 1000

So, rearranged, this is:
1 = 333 * 997 + (-332) * 1000

Therefore, comparing to the equation ax + by = 1, we have x = 333 and y = -332.

Therefore, the inverse of 997 modulo 1000 is 333. Wait, but let me check that. Let me compute 997 * 333 mod 1000.

But before I do that, let me make sure I didn't make a mistake in the algebra. Let me verify the steps again.

Starting from the second equation:
1 = 997 - 332 * 3
Then, substitute 3 = 1000 - 997:
1 = 997 - 332*(1000 - 997)
= 997 - 332*1000 + 332*997
= (1 + 332)*997 - 332*1000
= 333*997 - 332*1000
Yes, that seems correct.

So, according to this, 333 is the coefficient for 997, so 333 is the inverse. But let me check with actual multiplication.

Compute 997 * 333. Let me compute that.

First, note that 997 is 1000 - 3. So, 997 * 333 = (1000 - 3) * 333 = 1000*333 - 3*333.

So, modulo 1000, the 1000*333 term is 0, so we have -3*333 mod 1000.

Compute -3*333: that is -999. Then, -999 mod 1000 is 1, since -999 + 1000 = 1.

Yes! So 997 * 333 ≡ 1 mod 1000. Therefore, 333 is indeed the inverse.

Wait, but let me check with another method just to be sure. Maybe using another approach?

Alternatively, since we are working modulo 1000, maybe we can find x such that 997x ≡ 1 mod 1000.

But 997 ≡ -3 mod 1000. Therefore, the equation becomes -3x ≡ 1 mod 1000. Which is equivalent to 3x ≡ -1 mod 1000, or 3x ≡ 999 mod 1000.

So solving 3x ≡ 999 mod 1000.

Divide both sides by gcd(3, 1000). Since gcd(3, 1000) is 1, we can find the inverse of 3 mod 1000 and multiply.

So, inverse of 3 mod 1000. Let me find that.

Find y such that 3y ≡ 1 mod 1000.

Again, using Extended Euclidean Algorithm.

Let me do that.

Find gcd(3, 1000):

1000 = 3 * 333 + 1
3 = 1 * 3 + 0

So gcd is 1. Then backtracking:

1 = 1000 - 3 * 333

Therefore, 1 = (-333)*3 + 1*1000. Therefore, inverse of 3 mod 1000 is -333 mod 1000.

-333 mod 1000 is 1000 - 333 = 667.

Therefore, inverse of 3 is 667.

Therefore, going back to 3x ≡ 999 mod 1000. Multiply both sides by 667:

x ≡ 999 * 667 mod 1000.

Compute 999 * 667 mod 1000.

Again, note that 999 ≡ -1 mod 1000. Therefore, this is (-1)*667 mod 1000 = -667 mod 1000 = 1000 - 667 = 333.

So again, x ≡ 333 mod 1000. Therefore, same result.

Therefore, 333 is indeed the inverse.

Alternatively, maybe I can check with another approach. Let me try to compute 997 * x mod 1000 = 1.

But since 997 ≡ -3 mod 1000, so (-3)x ≡ 1 mod 1000. So 3x ≡ -1 mod 1000. Which is same as 3x ≡ 999 mod 1000.

Alternatively, maybe I can solve 3x ≡ 999 mod 1000.

Divide both sides by 3? Wait, but division in modular arithmetic is multiplication by inverse.

But since we already found that inverse of 3 is 667, so x ≡ 999 * 667 mod 1000.

But maybe compute 999 * 667 mod 1000.

Alternatively, since 999 ≡ -1, so (-1)*667 = -667 ≡ 333 mod 1000.

Same result.

Alternatively, maybe compute 3x ≡ 999 mod 1000. Let me divide both sides by something?

But since 3 and 1000 are coprime, we can divide.

But maybe I can solve for x by trial? Although that might take longer.

Alternatively, note that 3x ≡ 999 mod 1000. Let me divide both sides by gcd(3, 1000) = 1, so the equation is equivalent to x ≡ 999 * 3^{-1} mod 1000. Which is again 333.

Alternatively, maybe think of 3x ≡ 999 mod 1000. Let me write this as 3x - 1000k = 999 for some integer k.

So 3x - 1000k = 999.

Let me rearrange: 3x = 1000k + 999.

So x = (1000k + 999)/3.

We need x to be integer, so 1000k + 999 must be divisible by 3.

Compute 1000 mod 3. 1000 divided by 3 is 333*3 + 1, so 1000 ≡ 1 mod 3. Similarly, 999 is 3*333, so 999 ≡ 0 mod 3.

Therefore, 1000k + 999 ≡ 1*k + 0 ≡ k mod 3.

We need this to be 0 mod 3, so k ≡ 0 mod 3.

Therefore, let k = 3m for some integer m.

Then x = (1000*(3m) + 999)/3 = 1000m + 333.

Therefore, x ≡ 333 mod 1000.

Therefore, the smallest positive solution is 333.

So again, same answer.

Therefore, seems like 333 is the inverse.

But just to be thorough, let me compute 997 * 333 and check if it's 1 mod 1000.

Compute 997 * 333.

Well, 997 * 333. Let me compute this.

First, note that 1000 * 333 = 333,000.

Subtract 3 * 333 = 999.

Therefore, 997 * 333 = 333,000 - 999 = 332,001.

Now, 332,001 divided by 1000 is 332 with a remainder of 1. Therefore, 332,001 mod 1000 is 1.

Yes! So 997 * 333 ≡ 1 mod 1000.

Therefore, 333 is indeed the inverse.

Alternatively, maybe I can check with another number. Let me try multiplying 997 by 333 and see.

But since I already did that, and it worked, I think that's solid.

Alternatively, maybe check with another inverse? For example, if I think of 997 ≡ -3 mod 1000, so inverse of -3 is -inverse(3). Since inverse of 3 is 667, inverse of -3 is -667 mod 1000, which is 1000 - 667 = 333. Yep, same answer.

Therefore, all methods lead to 333.

Therefore, the answer is 333.

But just to make sure, let me try another approach. Maybe using the fact that 997 is congruent to -3 mod 1000, so solving (-3)x ≡ 1 mod 1000.

Which is equivalent to 3x ≡ -1 mod 1000.

So 3x ≡ 999 mod 1000.

Let me try to solve this equation step by step.

Let me write 3x = 999 + 1000k for some integer k.

So x = (999 + 1000k)/3.

We need x to be integer.

So 999 + 1000k must be divisible by 3.

As before, 999 is divisible by 3 (since 9+9+9=27, which is divisible by 3). 1000k mod 3 is (1*k) mod 3, since 1000 ≡ 1 mod 3.

Therefore, 999 + 1000k ≡ 0 + k ≡ k mod 3.

Therefore, k must be ≡ 0 mod 3.

So let k = 3m.

Then x = (999 + 1000*(3m))/3 = 999/3 + (1000*3m)/3 = 333 + 1000m.

Therefore, x ≡ 333 mod 1000.

Same result.

Therefore, regardless of the method, the inverse is 333.

Alternatively, maybe I can use the fact that 997 is congruent to -3 mod 1000, so inverse is -1/3 mod 1000. Which is -667 mod 1000, which is 333.

So, all methods converge.

Therefore, I can be confident that the inverse of 997 modulo 1000 is 333.

But just to make sure, let me check with another value. Let me pick a number close to 333 and see.

For example, let me try 333.

Compute 997 * 333.

As before, 997 * 333 = 332,001.

Divide by 1000: 332,001 / 1000 = 332.001. So the remainder is 1.

Yes, so 332,001 mod 1000 is 1.

Alternatively, if I take 333 + 1000 = 1333. Then 997 * 1333 would be 997*(1000 + 333) = 997*1000 + 997*333. Which is 997000 + 332001 = 1,329,001. Then mod 1000, that's 1. So same result.

Therefore, 333 is the correct inverse.

Alternatively, maybe check with another number. Let me try 333 - 1000 = -667. Let me check if -667 is also an inverse.

Compute 997 * (-667) mod 1000.

Again, 997 ≡ -3, so (-3)*(-667) = 2001. 2001 mod 1000 is 1.

Yes! So -667 is also an inverse, but since we need the answer between 0 and 999, we take -667 mod 1000, which is 333.

Therefore, same answer.

Therefore, I think I can be confident that 333 is the correct inverse.

Is there any other way to check? Maybe using another modulus?

Alternatively, since 1000 = 8 * 125, and 997 is congruent to -3 mod 8 and mod 125. Maybe use Chinese Remainder Theorem?

Let me try that approach.

Find x such that 997x ≡ 1 mod 8 and 997x ≡ 1 mod 125. Then combine the solutions.

First, compute 997 mod 8.

Divide 997 by 8: 8*124 = 992, so 997 - 992 = 5. Therefore, 997 ≡ 5 mod 8.

So equation 1: 5x ≡ 1 mod 8.

Solve for x.

Multiply both sides by inverse of 5 mod 8.

What is inverse of 5 mod 8?

Find y such that 5y ≡ 1 mod 8.

Testing y=5: 5*5=25≡1 mod8. Yes. So inverse is 5.

Therefore, x ≡ 5*1 ≡ 5 mod 8.

So x ≡ 5 mod 8.

Now, equation 2: 997x ≡ 1 mod 125.

Compute 997 mod 125.

Divide 997 by 125. 125*7 = 875. 997 - 875 = 122. So 997 ≡ 122 mod 125.

So equation: 122x ≡ 1 mod 125.

Need to solve for x.

Find inverse of 122 mod 125.

Note that 122 ≡ -3 mod 125. So inverse of -3 mod 125.

Which is same as -inverse(3) mod 125.

Find inverse of 3 mod 125.

Find y such that 3y ≡ 1 mod 125.

Use Extended Euclidean Algorithm.

Compute gcd(3, 125):

125 = 41*3 + 2
3 = 1*2 + 1
2 = 2*1 + 0

So gcd is 1. Now backtracking:

1 = 3 - 1*2
But 2 = 125 - 41*3, so substitute:

1 = 3 - 1*(125 - 41*3) = 3 - 125 + 41*3 = 42*3 - 125.

Therefore, 1 = 42*3 - 125*1.

Therefore, inverse of 3 mod 125 is 42.

Therefore, inverse of -3 mod 125 is -42 mod 125.

-42 mod 125 is 125 - 42 = 83.

Therefore, inverse of 122 mod 125 is 83.

Therefore, x ≡ 83 mod 125.

So now, we have:

x ≡ 5 mod 8
x ≡ 83 mod 125

We need to find x such that these two congruences hold. Then x will be the inverse mod 1000.

Let me solve this system.

Let x = 125k + 83 for some integer k.

Substitute into first equation: 125k + 83 ≡ 5 mod 8.

Compute 125 mod 8: 125 / 8 = 15*8 + 5, so 125 ≡ 5 mod 8.

Similarly, 83 mod 8: 83 / 8 = 10*8 + 3, so 83 ≡ 3 mod 8.

Therefore, equation becomes: 5k + 3 ≡ 5 mod 8
Subtract 3: 5k ≡ 2 mod 8
Solve for k: 5k ≡ 2 mod 8.

Multiply both sides by inverse of 5 mod 8, which we know is 5.

So k ≡ 2*5 ≡ 10 ≡ 2 mod 8.

Therefore, k = 8m + 2 for some integer m.

Therefore, x = 125*(8m + 2) + 83 = 1000m + 250 + 83 = 1000m + 333.

Therefore, x ≡ 333 mod 1000.

Therefore, the solution is 333 mod 1000.

Therefore, again, same answer.

Therefore, using Chinese Remainder Theorem also gives 333.

Therefore, this confirms that 333 is indeed the inverse.

So, after multiple methods: Extended Euclidean Algorithm, solving via modular inverses, Chinese Remainder Theorem, direct computation, all lead to the same answer.

Therefore, I can be confident that the inverse of 997 modulo 1000 is 333.

**Final Answer**
The inverse of $997$ modulo $1000$ is $\boxed{333}$.

To find the modular inverse of $ 997 $ modulo $ 1000 $, we are looking for an integer $ x $ such that:

$$
997x \equiv 1 \pmod{1000}
$$

---

### Step 1: Confirm that an inverse exists

An inverse exists if and only if the number and the modulus are coprime. That is, $ \gcd(997, 1000) = 1 $.

We compute:

$$
\gcd(997, 1000) = \gcd(997, 1000 - 997) = \gcd(997, 3)
$$
$$
\gcd(997, 3) = \gcd(3, 997 \mod 3) = \gcd(3, 1) = 1
$$

So, $ 997 $ and $ 1000 $ are coprime, and an inverse exists.

---

### Step 2: Use the Extended Euclidean Algorithm

We apply the Extended Euclidean Algorithm to find integers $ x $ and $ y $ such that:

$$
997x + 1000y = 1
$$

We perform the Euclidean algorithm:

$$
\begin{align*}
1000 &= 1 \cdot 997 + 3 \\
997 &= 332 \cdot 3 + 1 \\
3 &= 3 \cdot 1 + 0
\end{align*}
$$

Now, we backtrack to express 1 as a linear combination of 997 and 1000:

$$
1 = 997 - 332 \cdot 3
$$
$$
\text{But } 3 = 1000 - 1 \cdot 997
$$
$$
\Rightarrow 1 = 997 - 332 \cdot (1000 - 997) = 997 - 332 \cdot 1000 + 332 \cdot 997
$$
$$
\Rightarrow 1 = (1 + 332) \cdot 997 - 332 \cdot 1000 = 333 \cdot 997 - 332 \cdot 1000
$$

So, the coefficient of 997 is $ x = 333 $, which is the modular inverse of 997 modulo 1000.

---

### Step 3: Verify the result

We verify that:

$$
997 \cdot 333 \equiv 1 \pmod{1000}
$$

Note that $ 997 \equiv -3 \pmod{1000} $, so:

$$
997 \cdot 333 \equiv (-3) \cdot 333 = -999 \equiv 1 \pmod{1000}
$$

This confirms that $ 333 $ is indeed the inverse.

---

### Final Answer

$$
\boxed{333}
$$

Solution: We note that $$997\equiv -3\pmod{1000},$$and $$(-3)\cdot 333 = -999 = -1000+1\equiv 1\pmod{1000}.$$Therefore, $$997\cdot 333\equiv 1\pmod{1000},$$and the inverse of $997$ modulo $1000$ is $\boxed{333}$.


Explanation:
Okay, so I need to convert the repeating decimal 0.133113311331... into a fraction. The question says it's a repeating decimal, right? The bar is over 1331, so that means the digits 1331 repeat indefinitely. Let me recall how to convert repeating decimals to fractions.

I think the standard method involves setting the decimal equal to a variable, multiplying both sides by a power of 10 to shift the decimal point so that the repeating parts align, and then subtracting the equations to eliminate the repeating part. Let me try that.

Let me denote x = 0.133113311331...

First, I need to figure out how many digits are in the repeating part. The repeating part is 1331, which has 4 digits. So, if I multiply x by 10^4, which is 10000, that should shift the decimal point four places to the right. Let me write that down:

x = 0.133113311331...

10000x = 1331.133113311331...

Now, if I subtract the original x from this equation, the repeating parts should cancel out. Let me do that:

10000x - x = 1331.13311331... - 0.13311331...

So, 9999x = 1331

Therefore, x = 1331 / 9999

Wait, is that right? Let me check. If I subtract 0.1331... from 1331.1331..., the decimal parts should cancel, leaving 1331. So yes, 10000x - x = 1331. So x = 1331 / 9999.

But now I need to simplify this fraction. The question says that a and b are coprime, so I need to find the greatest common divisor (GCD) of 1331 and 9999 and divide both numerator and denominator by that GCD.

First, let me factor both numbers. Let me start with 1331. Hmm, 1331... I think that's 11 cubed. Let me check: 11*11 is 121, 121*11 is 1331. Yes, so 1331 = 11^3.

Now, let me factor 9999. Let me recall that 9999 is 10000 - 1. Which factors into... Well, 9999 = 9 * 1111. Wait, 9*1111 is 9999. Then 1111... Hmm, 1111. Let me check if 11 divides into 1111. 11*101 is 1111. Let me verify: 11*100 is 1100, plus 11 is 1111. Yes! So 1111 = 11*101. Therefore, 9999 = 9 * 11 * 101.

So, breaking it down:

9999 = 9 * 11 * 101 = 3^2 * 11 * 101

And 1331 is 11^3.

So, the GCD of 1331 and 9999 is the product of the common prime factors with the lowest exponents. The common prime factor is 11. In 1331, it's 11^3, and in 9999, it's 11^1. So the GCD is 11^1 = 11.

Therefore, we can divide numerator and denominator by 11:

1331 ÷ 11 = 121 (since 11*121 = 1331)

9999 ÷ 11 = 909 (since 9999 ÷ 11: Let me compute that. 11*900 = 9900, so 9999 - 9900 = 99, which is 11*9. So total is 900 + 9 = 909. So yes, 909.)

So the simplified fraction is 121/909. Wait, but let me check if 121 and 909 have any common factors.

121 is 11 squared. Let me factor 909. Let me divide 909 by 3: 9 + 0 + 9 = 18, which is divisible by 3. So 909 ÷ 3 = 303. Then 303 ÷ 3 = 101. So 909 = 3^2 * 101.

So 121 is 11^2, and 909 is 3^2 * 101. There are no common factors between numerator and denominator. Because 11 is not a factor of 909 (since 909 ÷ 11 is 82.63...), and 3 and 101 are not factors of 121. Therefore, the fraction 121/909 is in simplest terms.

Therefore, a is 121 and b is 909. Then a + b is 121 + 909. Let me compute that. 121 + 900 is 1021, plus 9 is 1030. Wait, 121 + 909: 121 + 900 = 1021, then +9 = 1030.

But wait, let me check again. Maybe I made a mistake in simplifying? Let me check the GCD again.

Original fraction: 1331 / 9999. GCD(1331, 9999). Since 1331 is 11^3, and 9999 is 9*11*101. So GCD is 11. Dividing numerator and denominator by 11 gives 121 / 909. Then 121 is 11^2, and 909 factors into 3^2 * 101. So no common factors. Therefore, yes, 121/909 is reduced.

But let me check if 121 and 909 have any common factors. Let me check if 101 divides into 121? 101*1 is 101, 101*2 is 202, which is larger. So no. How about 3? 1+2+1=4, which is not divisible by 3. So 3 doesn't divide 121. 121 divided by 11 is 11, which we already considered. So yes, 121 and 909 are coprime? Wait, 909 is 3^2 * 101. 121 is 11^2. So no common prime factors. Therefore, yes, the fraction is reduced.

Therefore, a + b is 121 + 909 = 1030.

But wait, let me check my calculation again. Maybe I made a mistake in the initial step? Let me verify the value of 0.13311331... as 1331/9999. Let me check with another method.

Alternatively, maybe I can think of the decimal as 0.1331 repeating. Let me think of it as 0.13311331... So the repeating block is four digits. So the standard formula for repeating decimals is that if you have a repeating decimal with n digits, then the decimal is equal to the repeating block divided by 10^n - 1. So in this case, the repeating block is 1331, which is four digits, so it should be 1331 / 9999. Which is what I had before. So that seems correct.

Alternatively, maybe I can check with a calculator? Well, since I don't have a calculator here, but let me do some approximate decimal conversion. Let me compute 1331 divided by 9999. Let me see, 1331 / 9999. Well, 1331 is approximately 1/7.5? Wait, 1/7 is approximately 0.1428, 1/8 is 0.125. So 0.1331 is between 1/8 and 1/7. Let me compute 1331 / 9999. Let me divide numerator and denominator by 11: 121 / 909. Let me divide 121 by 909. Let me approximate that. 121 divided by 909. Let me think of 121/909 ≈ 0.133... Let me check 0.133 * 909. 0.1 * 909 = 90.9, 0.03 * 909 = 27.27, 0.003 * 909 = 2.727. Adding them up: 90.9 + 27.27 = 118.17 + 2.727 ≈ 120.897. Which is very close to 121. So that suggests that 121/909 ≈ 0.133... which matches the original decimal. Therefore, that seems correct.

Therefore, my answer is 121 + 909 = 1030.

But wait, let me check once more if there is a mistake in simplifying. Let me check if 1331 and 9999 have a GCD of 11. Let me use the Euclidean algorithm to compute GCD(1331, 9999).

The Euclidean algorithm says that GCD(a, b) = GCD(b, a mod b). So let me compute GCD(9999, 1331).

First, divide 9999 by 1331. How many times does 1331 go into 9999? Let me compute 1331 * 7 = 9317. Then 1331 * 7 = 9317. Subtract that from 9999: 9999 - 9317 = 682. So GCD(1331, 682).

Now compute GCD(1331, 682). Divide 1331 by 682. 682 * 1 = 682. Subtract from 1331: 1331 - 682 = 649. So GCD(682, 649).

Now GCD(682, 649). Divide 682 by 649. 649 * 1 = 649. Subtract: 682 - 649 = 33. So GCD(649, 33).

GCD(649, 33). Divide 649 by 33. 33*19 = 627. Subtract: 649 - 627 = 22. So GCD(33, 22).

GCD(33, 22). Divide 33 by 22: 22*1 = 22. Subtract: 33 - 22 = 11. GCD(22, 11).

GCD(22, 11) is 11. Therefore, the GCD of 1331 and 9999 is 11. Which confirms my previous result. Therefore, dividing numerator and denominator by 11 gives 121/909, which is reduced.

Therefore, the answer is 121 + 909 = 1030.

But wait, just to be thorough, let me check if 121 and 909 have any common factors. Let me check GCD(121, 909). Using the Euclidean algorithm again.

GCD(909, 121). Divide 909 by 121. 121*7 = 847. Subtract: 909 - 847 = 62. So GCD(121, 62).

GCD(121, 62). Divide 121 by 62: 62*1 = 62. Subtract: 121 - 62 = 59. GCD(62, 59).

GCD(62, 59). Divide 62 by 59: 59*1 = 59. Subtract: 62 - 59 = 3. GCD(59, 3).

GCD(59, 3). Divide 59 by 3: 3*19 = 57. Subtract: 59 - 57 = 2. GCD(3, 2).

GCD(3, 2). Divide 3 by 2: 2*1 = 2. Subtract: 3 - 2 = 1. GCD(2, 1).

GCD(2, 1) is 1. Therefore, GCD(121, 909) is 1. Therefore, the fraction is indeed reduced completely.

Therefore, the answer is 121 + 909 = 1030.

But just to be 100% sure, let me check if there's another way to approach this problem. Maybe breaking down the decimal into parts? For example, 0.13311331... can be thought of as 0.1331 + 0.00001331 + 0.000000001331 + ... which is a geometric series.

Let me try that approach. Let me denote the decimal as x = 0.133113311331...

So, x = 0.1331 + 0.00001331 + 0.000000001331 + ...

This is a geometric series where the first term a = 0.1331 and the common ratio r = 0.0001. Because each term is multiplied by 0.0001 to get the next term.

So, the sum of an infinite geometric series is a / (1 - r). Therefore, x = 0.1331 / (1 - 0.0001) = 0.1331 / 0.9999.

But 0.1331 is 1331/10000, so substituting that in, x = (1331/10000) / (0.9999). But 0.9999 is 9999/10000. Therefore, dividing by 9999/10000 is the same as multiplying by 10000/9999. Therefore, x = (1331/10000) * (10000/9999) = 1331 / 9999. Which is the same result as before. So that confirms that x = 1331/9999.

Therefore, the same fraction as before. So again, simplifying gives 121/909, so a + b = 1030.

Therefore, I think my answer is correct.

But just to make sure, let me check if 121/909 is indeed equal to 0.13311331... Let me perform the division 121 ÷ 909.

Well, 121 divided by 909. Since 121 is less than 909, it's 0. Let me compute 121.0000 divided by 909.

First, 909 goes into 1210 once (since 909*1=909). Subtract 909 from 1210: 1210 - 909 = 301. Bring down a zero: 3010.

909 goes into 3010 three times (909*3=2727). Subtract: 3010 - 2727 = 283. Bring down a zero: 2830.

909 goes into 2830 three times (909*3=2727). Subtract: 2830 - 2727 = 103. Bring down a zero: 1030.

909 goes into 1030 once (909*1=909). Subtract: 1030 - 909 = 121. Bring down a zero: 1210.

Wait, now we have 1210 again, which was the first step. So the decimal repeats.

So the decimal expansion is 0.1331... Let me write down the decimal places:

First division step: 1 (after decimal), remainder 301. Then 3, remainder 283. Then 3, remainder 103. Then 1, remainder 121. Then it repeats.

So the decimal is 0.1331... So the decimal expansion is 0.13311331..., which matches the original decimal. Therefore, 121/909 is indeed equal to 0.13311331..., so the fraction is correct.

Therefore, the answer is 121 + 909 = 1030.

I think that's thorough. I considered multiple methods: the standard repeating decimal conversion, the geometric series approach, and verifying the decimal expansion. All lead to the same result. So I feel confident that the answer is 1030.

**Final Answer**
\boxed{1030}

To convert the repeating decimal $ 0.\overline{1331} $ into a fraction, we follow a standard method for converting repeating decimals to fractions.

---

### Step 1: Let the repeating decimal be a variable

Let:
$$
x = 0.\overline{1331}
$$

This means the repeating block is **1331**, which has **4 digits**.

---

### Step 2: Multiply to shift the decimal

To align the repeating parts, multiply both sides by $ 10^4 = 10000 $:

$$
10000x = 1331.\overline{1331}
$$

Now subtract the original equation from this:

$$
10000x - x = 1331.\overline{1331} - 0.\overline{1331}
$$

$$
9999x = 1331
$$

$$
x = \frac{1331}{9999}
$$

---

### Step 3: Simplify the fraction

We now simplify $ \frac{1331}{9999} $ by finding the greatest common divisor (GCD) of the numerator and denominator.

#### Prime factorization:

- $ 1331 = 11^3 $
- $ 9999 = 9 \times 1111 = 9 \times 11 \times 101 = 3^2 \times 11 \times 101 $

So the GCD of 1331 and 9999 is $ 11 $.

Divide both numerator and denominator by 11:

$$
\frac{1331}{9999} = \frac{1331 \div 11}{9999 \div 11} = \frac{121}{909}
$$

Now check if $ \frac{121}{909} $ is in lowest terms.

- $ 121 = 11^2 $
- $ 909 = 3^2 \times 101 $

There are no common prime factors between numerator and denominator, so the fraction is fully reduced.

---

### Step 4: Compute $ a + b $

The simplified fraction is $ \frac{121}{909} $, so:

$$
a = 121, \quad b = 909
$$

$$
a + b = 121 + 909 = \boxed{1030}
$$

---

### Final Answer

$$
\boxed{1030}
$$

[Assistant]
Explanation:
Okay, so I need to convert the repeating decimal 0.133113311331... into a fraction. The question says it's a repeating decimal, right? The bar is over 1331, so that means the digits 1331 repeat indefinitely. Let me recall how to convert repeating decimals to fractions.

I think the standard method involves setting the decimal equal to a variable, multiplying both sides by a power of 10 to shift the decimal point so that the repeating parts align, and then subtracting the equations to eliminate the repeating part. Let me try that.

Let me denote x = 0.133113311331...

First, I need to figure out how many digits are in the repeating part. The repeating part is 1331, which has 4 digits. So, if I multiply x by 10^4, which is 10000, that should shift the decimal point four places to the right. Let me write that down:

x = 0.133113311331...

10000x = 1331.133113311331...

Now, if I subtract the original x from this equation, the repeating parts should cancel out. Let me do that:

10000x - x = 1331.13311331... - 0.13311331...

So, 9999x = 1331

Therefore, x = 1331 / 9999

Wait, is that right? Let me check. If I subtract 0.1331... from 1331.1331..., the decimal parts should cancel, leaving 1331. So yes, 10000x - x = 1331. So x = 1331 / 9999.

But now I need to simplify this fraction. The question says that a and b are coprime, so I need to find the greatest common divisor (GCD) of 1331 and 9999 and divide both numerator and denominator by that GCD.

First, let me factor both numbers. Let me start with 1331. Hmm, 1331... I think that's 11 cubed. Let me check: 11*11 is 121, 121*11 is 1331. Yes, so 1331 = 11^3.

Now, let me factor 9999. Let me recall that 9999 is 10000 - 1. Which factors into... Well, 9999 = 9 * 1111. Wait, 9*1111 is 9999. Then 1111... Hmm, 1111. Let me check if 11 divides into 1111. 11*101 is 1111. Let me verify: 11*100 is 1100, plus 11 is 1111. Yes! So 1111 = 11*101. Therefore, 9999 = 9 * 11 * 101.

So, breaking it down:

9999 = 9 * 11 * 101 = 3^2 * 11 * 101

And 1331 is 11^3.

So, the GCD of 1331 and 9999 is the product of the common prime factors with the lowest exponents. The common prime factor is 11. In 1331, it's 11^3, and in 9999, it's 11^1. So the GCD is 11^1 = 11.

Therefore, we can divide numerator and denominator by 11:

1331 ÷ 11 = 121 (since 11*121 = 1331)

9999 ÷ 11 = 909 (since 9999 ÷ 11: Let me compute that. 11*900 = 9900, so 9999 - 9900 = 99, which is 11*9. So total is 900 + 9 = 909. So yes, 909.)

So the simplified fraction is 121/909. Wait, but let me check if 121 and 909 have any common factors.

121 is 11 squared. Let me factor 909. Let me divide 909 by 3: 9 + 0 + 9 = 18, which is divisible by 3. So 909 ÷ 3 = 303. Then 303 ÷ 3 = 101. So 909 = 3^2 * 101.

So 121 is 11^2, and 909 is 3^2 * 101. There are no common factors between numerator and denominator. Because 11 is not a factor of 909 (since 909 ÷ 11 is 82.63...), and 3 and 101 are not factors of 121. Therefore, the fraction 121/909 is in simplest terms.

Therefore, a is 121 and b is 909. Then a + b is 121 + 909. Let me compute that. 121 + 900 is 1021, plus 9 is 1030. Wait, 121 + 909: 121 + 900 = 1021, then +9 = 1030.

But wait, let me check again. Maybe I made a mistake in simplifying? Let me check the GCD again.

Original fraction: 1331 / 9999. GCD(1331, 9999). Since 1331 is 11^3, and 9999 is 9*11*101. So GCD is 11. Dividing numerator and denominator by 11 gives 121 / 909. Then 121 is 11^2, and 909 factors into 3^2 * 101. So no common factors. Therefore, yes, 121/909 is reduced.

But let me check if 121 and 909 have any common factors. Let me check if 101 divides into 121? 101*1 is 101, 101*2 is 202, which is larger. So no. How about 3? 1+2+1=4, which is not divisible by 3. So 3 doesn't divide 121. 121 divided by 11 is 11, which we already considered. So yes, 121 and 909 are coprime? Wait, 909 is 3^2 * 101. 121 is 11^2. So no common prime factors. Therefore, yes, the fraction is reduced.

Therefore, a + b is 121 + 909 = 1030.

But wait, let me check my calculation again. Maybe I made a mistake in the initial step? Let me verify the value of 0.13311331... as 1331/9999. Let me check with another method.

Alternatively, maybe I can think of the decimal as 0.1331 repeating. Let me think of it as 0.13311331... So the repeating block is four digits. So the standard formula for repeating decimals is that if you have a repeating decimal with n digits, then the decimal is equal to the repeating block divided by 10^n - 1. So in this case, the repeating block is 1331, which is four digits, so it should be 1331 / 9999. Which is what I had before. So that seems correct.

Alternatively, maybe I can check with a calculator? Well, since I don't have a calculator here, but let me do some approximate decimal conversion. Let me compute 1331 divided by 9999. Let me see, 1331 / 9999. Well, 1331 is approximately 1/7.5? Wait, 1/7 is approximately 0.1428, 1/8 is 0.125. So 0.1331 is between 1/8 and 1/7. Let me compute 1331 / 9999. Let me divide numerator and denominator by 11: 121 / 909. Let me divide 121 by 909. Let me approximate that. 121 divided by 909. Let me think of 121/909 ≈ 0.133... Let me check 0.133 * 909. 0.1 * 909 = 90.9, 0.03 * 909 = 27.27, 0.003 * 909 = 2.727. Adding them up: 90.9 + 27.27 = 118.17 + 2.727 ≈ 120.897. Which is very close to 121. So that suggests that 121/909 ≈ 0.133... which matches the original decimal. Therefore, that seems correct.

Therefore, my answer is 121 + 909 = 1030.

But wait, let me check once more if there is a mistake in simplifying. Let me check if 1331 and 9999 have a GCD of 11. Let me use the Euclidean algorithm to compute GCD(1331, 9999).

The Euclidean algorithm says that GCD(a, b) = GCD(b, a mod b). So let me compute GCD(9999, 1331).

First, divide 9999 by 1331. How many times does 1331 go into 9999? Let me compute 1331 * 7 = 9317. Then 1331 * 7 = 9317. Subtract that from 9999: 9999 - 9317 = 682. So GCD(1331, 682).

Now compute GCD(1331, 682). Divide 1331 by 682. 682 * 1 = 682. Subtract from 1331: 1331 - 682 = 649. So GCD(682, 649).

Now GCD(682, 649). Divide 682 by 649. 649 * 1 = 649. Subtract: 682 - 649 = 33. So GCD(649, 33).

GCD(649, 33). Divide 649 by 33. 33*19 = 627. Subtract: 649 - 627 = 22. So GCD(33, 22).

GCD(33, 22). Divide 33 by 22: 22*1 = 22. Subtract: 33 - 22 = 11. GCD(22, 11).

GCD(22, 11) is 11. Therefore, the GCD of 1331 and 9999 is 11. Which confirms my previous result. Therefore, dividing numerator and denominator by 11 gives 121/909, which is reduced.

Therefore, the answer is 121 + 909 = 1030.

But wait, just to be thorough, let me check if 121 and 909 have any common factors. Let me check GCD(121, 909). Using the Euclidean algorithm again.

GCD(909, 121). Divide 909 by 121. 121*7 = 847. Subtract: 909 - 847 = 62. So GCD(121, 62).

GCD(121, 62). Divide 121 by 62: 62*1 = 62. Subtract: 121 - 62 = 59. GCD(62, 59).

GCD(62, 59). Divide 62 by 59: 59*1 = 59. Subtract: 62 - 59 = 3. GCD(59, 3).

GCD(59, 3). Divide 59 by 3: 3*19 = 57. Subtract: 59 - 57 = 2. GCD(3, 2).

GCD(3, 2). Divide 3 by 2: 2*1 = 2. Subtract: 3 - 2 = 1. GCD(2, 1).

GCD(2, 1) is 1. Therefore, GCD(121, 909) is 1. Therefore, the fraction is indeed reduced completely.

Therefore, the answer is 121 + 909 = 1030.

But just to be 100% sure, let me check if there's another way to approach this problem. Maybe breaking down the decimal into parts? For example, 0.13311331... can be thought of as 0.1331 + 0.00001331 + 0.000000001331 + ... which is a geometric series.

Let me try that approach. Let me denote the decimal as x = 0.133113311331...

So, x = 0.1331 + 0.00001331 + 0.000000001331 + ...

This is a geometric series where the first term a = 0.1331 and the common ratio r = 0.0001. Because each term is multiplied by 0.0001 to get the next term.

So, the sum of an infinite geometric series is a / (1 - r). Therefore, x = 0.1331 / (1 - 0.0001) = 0.1331 / 0.9999.

But 0.1331 is 1331/10000, so substituting that in, x = (1331/10000) / (0.9999). But 0.9999 is 9999/10000. Therefore, dividing by 9999/10000 is the same as multiplying by 10000/9999. Therefore, x = (1331/10000) * (10000/9999) = 1331 / 9999. Which is the same result as before. So that confirms that x = 1331/9999.

Therefore, the same fraction as before. So again, simplifying gives 121/909, so a + b = 1030.

Therefore, I think my answer is correct.

But just to make sure, let me check if 121/909 is indeed equal to 0.13311331... Let me perform the division 121 ÷ 909.

Well, 121 divided by 909. Since 121 is less than 909, it's 0. Let me compute 121.0000 divided by 909.

First, 909 goes into 1210 once (since 909*1=909). Subtract 909 from 1210: 1210 - 909 = 301. Bring down a zero: 3010.

909 goes into 3010 three times (909*3=2727). Subtract: 3010 - 2727 = 283. Bring down a zero: 2830.

909 goes into 2830 three times (909*3=2727). Subtract: 2830 - 2727 = 103. Bring down a zero: 1030.

909 goes into 1030 once (909*1=909). Subtract: 1030 - 909 = 121. Bring down a zero: 1210.

Wait, now we have 1210 again, which was the first step. So the decimal repeats.

So the decimal expansion is 0.1331... Let me write down the decimal places:

First division step: 1 (after decimal), remainder 301. Then 3, remainder 283. Then 3, remainder 103. Then 1, remainder 121. Then it repeats.

So the decimal is 0.1331... So the decimal expansion is 0.13311331..., which matches the original decimal. Therefore, 121/909 is indeed equal to 0.13311331..., so the fraction is correct.

Therefore, the answer is 121 + 909 = 1030.

I think that's thorough. I considered multiple methods: the standard repeating decimal conversion, the geometric series approach, and verifying the decimal expansion. All lead to the same result. So I feel confident that the answer is 1030.

**Final Answer**
\boxed{1030}

To convert the repeating decimal $ 0.\overline{1331} $ into a fraction, we follow a standard method for converting repeating decimals to fractions.

---

### Step 1: Let the repeating decimal be a variable

Let:
$$
x = 0.\overline{1331}
$$

This means the repeating block is **1331**, which has **4 digits**.

---

### Step 2: Multiply to shift the decimal

To align the repeating parts, multiply both sides by $ 10^4 = 10000 $:

$$
10000x = 1331.\overline{1331}
$$

Now subtract the original equation from this:

$$
10000x - x = 1331.\overline{1331} - 0.\overline{1331}
$$

$$
9999x = 1331
$$

$$
x = \frac{1331}{9999}
$$

---

### Step 3: Simplify the fraction

We now simplify $ \frac{1331}{9999} $ by finding the greatest common divisor (GCD) of the numerator and denominator.

#### Prime factorization:

- $ 1331 = 11^3 $
- $ 9999 = 9 \times 1111 = 9 \times 11 \times 101 = 3^2 \times 11 \times 101 $

So the GCD of 1331 and 9999 is $ 11 $.

Divide both numerator and denominator by 11:

$$
\frac{1331}{9999} = \frac{1331 \div 11}{9999 \div 11} = \frac{121}{909}
$$

Now check if $ \frac{121}{909} $ is in lowest terms.

- $ 121 = 11^2 $
- $ 909 = 3^2 \times 101 $

There are no common prime factors between numerator and denominator, so the fraction is fully reduced.

---

### Step 4: Compute $ a + b $

The simplified fraction is $ \frac{121}{909} $, so:

$$
a = 121, \quad b = 909
$$

$$
a + b = 121 + 909 = \boxed{1030}
$$

---

### Final Answer

$$
\boxed{1030}
$$

Solution: Let $x = 0.\overline{1331}$, so $10000x = 1331.\overline{1331}$. As a result, $9999x = 1331$, so $x = \frac{1331}{9999}$. We can factor out 11 out of both the numerator and denominator, so $x = \frac{121}{909}$. Thus $a+b=121+909 = \boxed{1030}$.


Explanation:
Okay, so I need to figure out how many dollars are equivalent to 1,000,000 lire based on the exchange rate from 1992. Let me start by understanding the information given.

In 1992, a scoop of gelato in Italy cost 1200 lire, and the same gelato cost $1.50 in the U.S. So, they're telling me that 1200 lire is equivalent to $1.50. My job is to find out how much 1,000,000 lire would be in dollars using that exchange rate.

First, I think I need to find out the exchange rate between lire and dollars. That is, how many dollars is 1 lire worth? Or maybe how many lire is 1 dollar worth? Let me think.

Since 1200 lire equals $1.50, I can set up a ratio or a proportion to find the exchange rate. Let me write that down.

If 1200 lire = $1.50, then 1 lire would be equal to $1.50 divided by 1200. Let me calculate that.

So, $1.50 divided by 1200 lire. Let me do the division.

First, maybe convert $1.50 to cents to make it easier? Wait, maybe not necessary. Let me just do decimal division.

1.50 divided by 1200. Let me write that as 1.5 / 1200.

Alternatively, I can think of it as 1.5 divided by 1200. Let me compute that.

Well, 1.5 divided by 1200. Let me think of 1.5 divided by 12 is 0.125, right? Because 1.5 divided by 12 is the same as 1.5 divided by 12. So, 1.5 divided by 12 is 0.125. But here, it's divided by 1200, which is 12 times 100. So, if I divide by 12 first, I get 0.125, then divide by 100, which would be 0.00125.

So, 1 lire is equal to $0.00125? Let me check that.

If 1 lire is $0.00125, then 1200 lire would be 1200 * 0.00125. Let me compute that.

1200 * 0.00125. Well, 1200 * 0.001 is 1.2, and 1200 * 0.00025 is 0.3. So adding them together, 1.2 + 0.3 = 1.5. Yes, that's correct. So that checks out.

So, the exchange rate is 1 lire = $0.00125.

Alternatively, maybe I can write this as a fraction. $1.50 / 1200 lire. So, simplifying that fraction.

Let me write it as 1.50 / 1200. To make it easier, multiply numerator and denominator by 100 to eliminate decimals. That would be 150 / 120000. Simplify that fraction.

Divide numerator and denominator by 150. 150 divided by 150 is 1. 120000 divided by 150 is... let me calculate that. 120000 divided by 150. Well, 150 times 800 is 120,000. So, 120000 / 150 = 800. So, the simplified fraction is 1 / 800.

So, 1 lire is equal to 1/800 dollars. Which is 0.00125 dollars. Yep, same as before.

So, that means that 1 lire = $1/800. Therefore, to find out how much 1,000,000 lire is in dollars, I can multiply 1,000,000 by 1/800.

Let me compute that.

1,000,000 divided by 800. Let me do this division.

First, note that 1,000,000 divided by 800. Let me simplify.

Divide numerator and denominator by 100: 10,000 / 8.

Because 1,000,000 divided by 100 is 10,000, and 800 divided by 100 is 8.

So, 10,000 divided by 8.

Well, 10,000 divided by 8. Let me compute that.

8 goes into 10 once, remainder 2. Bring down the 0: 20. 8 into 20 is 2, remainder 4. Bring down the next 0: 40. 8 into 40 is 5. Bring down the next 0: 0. So, the result is 1250.

Wait, 8 * 1250 = 10,000. Let me check: 8 * 1000 = 8000, 8 * 250 = 2000, so 8000 + 2000 = 10,000. Yes, correct.

Therefore, 1,000,000 lire divided by 800 is 1250 dollars.

So, 1,000,000 lire is equivalent to $1250.

Wait, let me just verify this with another approach to make sure I didn't make a mistake.

If 1200 lire is $1.50, then how much is 1,000,000 lire?

We can set up a proportion.

Let me write that as:

1200 lire / $1.50 = 1,000,000 lire / x dollars

So, solving for x.

Cross-multiplying, 1200 * x = 1,000,000 * 1.50

Therefore, x = (1,000,000 * 1.50) / 1200

Let me compute that.

First, compute 1,000,000 * 1.50. That's 1,500,000.

Then divide by 1200.

So, 1,500,000 / 1200.

Again, simplify. Divide numerator and denominator by 100: 15,000 / 12.

Divide numerator and denominator by 6: 15,000 / 6 = 2,500; 12 / 6 = 2. So, 2,500 / 2 = 1,250.

So, x = 1,250.

Same result. So that's consistent.

Therefore, 1,000,000 lire is equivalent to $1,250.

Alternatively, another way to think about it:

If 1200 lire = $1.50, then how many sets of 1200 lire are in 1,000,000 lire?

That would be 1,000,000 divided by 1200.

Then, multiply that number by $1.50 to get the dollar amount.

So, let's compute that.

First, 1,000,000 / 1200. Let me compute that.

Again, divide numerator and denominator by 100: 10,000 / 12.

Which is approximately 833.333...

Then multiply by $1.50.

So, 833.333... * 1.50.

Let me compute that.

Well, 833.333... * 1.5 is the same as 833.333... * 3/2.

Which is (833.333... * 3) / 2.

Calculating 833.333... * 3:

Well, 800 * 3 = 2400, 33.333... * 3 = 100. So total is 2400 + 100 = 2500.

Then divide by 2: 2500 / 2 = 1250.

Same answer again.

So, all methods lead to $1,250.

Therefore, I can be confident that the answer is $1,250.

But just to recap, the key steps were:

1. Find the exchange rate by dividing the dollar amount by the number of lire: $1.50 / 1200 lire = $0.00125 per lire.

2. Multiply 1,000,000 lire by $0.00125 to get the dollar amount: 1,000,000 * 0.00125 = 1,250.

Alternatively, using proportions or unit conversion, all methods confirm the same result.

I think that's thorough. I don't see any mistakes in my reasoning. All the different approaches cross-validate each other. So, I think the answer is definitely $1,250.

**Final Answer**
The equivalent amount in dollars is \boxed{1250}.

To determine how many dollars are equivalent to 1,000,000 lire based on the given exchange rate, we start by analyzing the provided information:

- In 1992, a scoop of gelato cost **1200 lire** in Italy and **$1.50** in the U.S.
- This implies that **1200 lire = $1.50**.

---

### Step 1: Determine the Exchange Rate

We want to find how many dollars one lire is worth. This is done by dividing the dollar amount by the number of lire:

$$
\text{Exchange rate} = \frac{1.50}{1200} = 0.00125 \text{ dollars per lire}
$$

So, **1 lire = $0.00125**.

---

### Step 2: Convert 1,000,000 Lire to Dollars

Now, to find the dollar equivalent of 1,000,000 lire, we multiply:

$$
1,000,000 \times 0.00125 = 1250
$$

---

### Step 3: Verification via Proportion

We can also verify this using a proportion:

$$
\frac{1200 \text{ lire}}{1.50 \text{ dollars}} = \frac{1,000,000 \text{ lire}}{x \text{ dollars}}
$$

Cross-multiplying:

$$
1200x = 1,000,000 \times 1.50 \Rightarrow x = \frac{1,500,000}{1200} = 1250
$$

This confirms the result.

---

### Final Answer

$$
\boxed{1250}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many dollars are equivalent to 1,000,000 lire based on the exchange rate from 1992. Let me start by understanding the information given.

In 1992, a scoop of gelato in Italy cost 1200 lire, and the same gelato cost $1.50 in the U.S. So, they're telling me that 1200 lire is equivalent to $1.50. My job is to find out how much 1,000,000 lire would be in dollars using that exchange rate.

First, I think I need to find out the exchange rate between lire and dollars. That is, how many dollars is 1 lire worth? Or maybe how many lire is 1 dollar worth? Let me think.

Since 1200 lire equals $1.50, I can set up a ratio or a proportion to find the exchange rate. Let me write that down.

If 1200 lire = $1.50, then 1 lire would be equal to $1.50 divided by 1200. Let me calculate that.

So, $1.50 divided by 1200 lire. Let me do the division.

First, maybe convert $1.50 to cents to make it easier? Wait, maybe not necessary. Let me just do decimal division.

1.50 divided by 1200. Let me write that as 1.5 / 1200.

Alternatively, I can think of it as 1.5 divided by 1200. Let me compute that.

Well, 1.5 divided by 1200. Let me think of 1.5 divided by 12 is 0.125, right? Because 1.5 divided by 12 is the same as 1.5 divided by 12. So, 1.5 divided by 12 is 0.125. But here, it's divided by 1200, which is 12 times 100. So, if I divide by 12 first, I get 0.125, then divide by 100, which would be 0.00125.

So, 1 lire is equal to $0.00125? Let me check that.

If 1 lire is $0.00125, then 1200 lire would be 1200 * 0.00125. Let me compute that.

1200 * 0.00125. Well, 1200 * 0.001 is 1.2, and 1200 * 0.00025 is 0.3. So adding them together, 1.2 + 0.3 = 1.5. Yes, that's correct. So that checks out.

So, the exchange rate is 1 lire = $0.00125.

Alternatively, maybe I can write this as a fraction. $1.50 / 1200 lire. So, simplifying that fraction.

Let me write it as 1.50 / 1200. To make it easier, multiply numerator and denominator by 100 to eliminate decimals. That would be 150 / 120000. Simplify that fraction.

Divide numerator and denominator by 150. 150 divided by 150 is 1. 120000 divided by 150 is... let me calculate that. 120000 divided by 150. Well, 150 times 800 is 120,000. So, 120000 / 150 = 800. So, the simplified fraction is 1 / 800.

So, 1 lire is equal to 1/800 dollars. Which is 0.00125 dollars. Yep, same as before.

So, that means that 1 lire = $1/800. Therefore, to find out how much 1,000,000 lire is in dollars, I can multiply 1,000,000 by 1/800.

Let me compute that.

1,000,000 divided by 800. Let me do this division.

First, note that 1,000,000 divided by 800. Let me simplify.

Divide numerator and denominator by 100: 10,000 / 8.

Because 1,000,000 divided by 100 is 10,000, and 800 divided by 100 is 8.

So, 10,000 divided by 8.

Well, 10,000 divided by 8. Let me compute that.

8 goes into 10 once, remainder 2. Bring down the 0: 20. 8 into 20 is 2, remainder 4. Bring down the next 0: 40. 8 into 40 is 5. Bring down the next 0: 0. So, the result is 1250.

Wait, 8 * 1250 = 10,000. Let me check: 8 * 1000 = 8000, 8 * 250 = 2000, so 8000 + 2000 = 10,000. Yes, correct.

Therefore, 1,000,000 lire divided by 800 is 1250 dollars.

So, 1,000,000 lire is equivalent to $1250.

Wait, let me just verify this with another approach to make sure I didn't make a mistake.

If 1200 lire is $1.50, then how much is 1,000,000 lire?

We can set up a proportion.

Let me write that as:

1200 lire / $1.50 = 1,000,000 lire / x dollars

So, solving for x.

Cross-multiplying, 1200 * x = 1,000,000 * 1.50

Therefore, x = (1,000,000 * 1.50) / 1200

Let me compute that.

First, compute 1,000,000 * 1.50. That's 1,500,000.

Then divide by 1200.

So, 1,500,000 / 1200.

Again, simplify. Divide numerator and denominator by 100: 15,000 / 12.

Divide numerator and denominator by 6: 15,000 / 6 = 2,500; 12 / 6 = 2. So, 2,500 / 2 = 1,250.

So, x = 1,250.

Same result. So that's consistent.

Therefore, 1,000,000 lire is equivalent to $1,250.

Alternatively, another way to think about it:

If 1200 lire = $1.50, then how many sets of 1200 lire are in 1,000,000 lire?

That would be 1,000,000 divided by 1200.

Then, multiply that number by $1.50 to get the dollar amount.

So, let's compute that.

First, 1,000,000 / 1200. Let me compute that.

Again, divide numerator and denominator by 100: 10,000 / 12.

Which is approximately 833.333...

Then multiply by $1.50.

So, 833.333... * 1.50.

Let me compute that.

Well, 833.333... * 1.5 is the same as 833.333... * 3/2.

Which is (833.333... * 3) / 2.

Calculating 833.333... * 3:

Well, 800 * 3 = 2400, 33.333... * 3 = 100. So total is 2400 + 100 = 2500.

Then divide by 2: 2500 / 2 = 1250.

Same answer again.

So, all methods lead to $1,250.

Therefore, I can be confident that the answer is $1,250.

But just to recap, the key steps were:

1. Find the exchange rate by dividing the dollar amount by the number of lire: $1.50 / 1200 lire = $0.00125 per lire.

2. Multiply 1,000,000 lire by $0.00125 to get the dollar amount: 1,000,000 * 0.00125 = 1,250.

Alternatively, using proportions or unit conversion, all methods confirm the same result.

I think that's thorough. I don't see any mistakes in my reasoning. All the different approaches cross-validate each other. So, I think the answer is definitely $1,250.

**Final Answer**
The equivalent amount in dollars is \boxed{1250}.

To determine how many dollars are equivalent to 1,000,000 lire based on the given exchange rate, we start by analyzing the provided information:

- In 1992, a scoop of gelato cost **1200 lire** in Italy and **$1.50** in the U.S.
- This implies that **1200 lire = $1.50**.

---

### Step 1: Determine the Exchange Rate

We want to find how many dollars one lire is worth. This is done by dividing the dollar amount by the number of lire:

$$
\text{Exchange rate} = \frac{1.50}{1200} = 0.00125 \text{ dollars per lire}
$$

So, **1 lire = $0.00125**.

---

### Step 2: Convert 1,000,000 Lire to Dollars

Now, to find the dollar equivalent of 1,000,000 lire, we multiply:

$$
1,000,000 \times 0.00125 = 1250
$$

---

### Step 3: Verification via Proportion

We can also verify this using a proportion:

$$
\frac{1200 \text{ lire}}{1.50 \text{ dollars}} = \frac{1,000,000 \text{ lire}}{x \text{ dollars}}
$$

Cross-multiplying:

$$
1200x = 1,000,000 \times 1.50 \Rightarrow x = \frac{1,500,000}{1200} = 1250
$$

This confirms the result.

---

### Final Answer

$$
\boxed{1250}
$$

Solution: Multiply both sides of $1200\text{ lire}=\$1.50$ by $1,\!000,\!000/1200$ to find that one million lire equals $\frac{3}{2}\cdot10,\!000/12=\boxed{1250}$ dollars.


Explanation:
Okay, so I need to figure out what the square root of 53 is in its simplest radical form. Hmm, let me start by recalling what simplest radical form means. From what I remember, simplest radical form for a square root means that the number under the square root (the radicand) doesn't have any perfect square factors other than 1. So, if I can factor 53 into a product where one of the factors is a perfect square, then I can simplify it. But if it doesn't have any perfect square factors besides 1, then it's already in its simplest form.

Alright, so first step: check if 53 is a perfect square. Let me think... The perfect squares around that number are 49 (which is 7 squared) and 64 (which is 8 squared). 53 is between those two. Since 7 squared is 49 and 8 squared is 64, 53 isn't a perfect square. So, that means it can't be simplified into an integer.

But maybe it can be broken down into a product of a square number and another number? Let me try factoring 53. Wait, 53 is a prime number, right? Let me check. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Let me test divisibility.

Is 53 divisible by 2? No, it's odd. Divisible by 3? Let's see, 5 + 3 = 8, and 8 isn't divisible by 3, so no. Divisible by 5? It doesn't end with 0 or 5, so no. Next prime is 7. 7 times 7 is 49, 7 times 8 is 56, which is more than 53. So 53 isn't divisible by 7. The next prime is 11, but 11 times 5 is 55, which is more than 53. So yeah, 53 is a prime number.

Since 53 is prime, its only factors are 1 and 53. Therefore, when we try to break down √53, we can't factor it into any numbers other than 1 and 53. Since 1 is a perfect square (1 squared is 1), but that doesn't help us simplify the radical. So, there are no perfect square factors other than 1.

Therefore, √53 cannot be simplified further. So, the simplest radical form of √53 is just √53.

Wait, but let me double-check. Maybe I made a mistake in assuming that 53 is prime? Let me try dividing 53 by some other numbers. For example, 53 divided by 13 is about 4.07, which isn't an integer. 17? 17 times 3 is 51, 17 times 4 is 68. Nope. 19? 19 times 2 is 38, 19 times 3 is 57. Nope. So, yeah, 53 is definitely a prime number.

Therefore, there are no square factors other than 1. So, √53 is already in its simplest radical form.

But just to make sure, let me recall the definition of simplest radical form again. It requires that the radicand has no perfect square factors other than 1, and that the index of the radical is as small as possible. Since the index here is 2 (square root), and there are no perfect square factors, it's already simplified.

Alternatively, if there was a factor like 4, 9, 16, etc., we could take that out of the square root. For example, √18 can be simplified to 3√2 because 18 = 9*2, and 9 is a perfect square. But in this case, since 53 is prime, there's no such factor.

Another way to think about it is that if you try to factor 53 into two numbers, say a and b, such that a*b = 53, then one of them has to be 1 and the other 53. Since 1 is a perfect square, but that doesn't help. So, there's no way to write 53 as a product of a perfect square and another number greater than 1.

Therefore, I think I can confidently say that √53 is already in its simplest radical form.

But just to be thorough, let me check if there's any other way to represent it. For example, sometimes people might think about rationalizing the denominator or something, but that's when you have a radical in the denominator. In this case, there's no denominator, so that doesn't apply.

Alternatively, could it be expressed as a sum or difference of square roots? Like, maybe √53 can be written as √a - √b or something? But that's not really simplifying it; that's more of an expansion. And I don't think that's necessary here.

Also, if I approximate √53, it's about 7.28, but that's a decimal approximation, not a radical form. The question specifically asks for simplest radical form, so decimal isn't the answer here.

So, after all that, I think the answer is just √53.

**Final Answer**
The simplest radical form of $\sqrt{53}$ is \boxed{\sqrt{53}}.

To determine the simplest radical form of $\sqrt{53}$, we begin by understanding what "simplest radical form" entails. A square root is in simplest radical form if:

1. The number under the square root (the **radicand**) has **no perfect square factors** other than 1.
2. The **index of the radical** (in this case, 2 for square roots) is as small as possible.

---

### Step 1: Check if 53 is a perfect square

We know that:
- $7^2 = 49$
- $8^2 = 64$

Since $53$ lies between $49$ and $64$, it is **not a perfect square**.

---

### Step 2: Factor 53 to check for perfect square factors

We now examine the prime factorization of 53. Since 53 is a **prime number**, its only factors are 1 and 53. This means:

- The only perfect square factor of 53 is $1$, which doesn't help in simplifying the square root.

---

### Step 3: Conclusion

Because 53 has no perfect square factors other than 1, the square root $\sqrt{53}$ **cannot be simplified further**. Therefore, it is already in its **simplest radical form**.

---

### Final Answer

$$
\boxed{\sqrt{53}}
$$

[Assistant]
Explanation:
Okay, so I need to figure out what the square root of 53 is in its simplest radical form. Hmm, let me start by recalling what simplest radical form means. From what I remember, simplest radical form for a square root means that the number under the square root (the radicand) doesn't have any perfect square factors other than 1. So, if I can factor 53 into a product where one of the factors is a perfect square, then I can simplify it. But if it doesn't have any perfect square factors besides 1, then it's already in its simplest form.

Alright, so first step: check if 53 is a perfect square. Let me think... The perfect squares around that number are 49 (which is 7 squared) and 64 (which is 8 squared). 53 is between those two. Since 7 squared is 49 and 8 squared is 64, 53 isn't a perfect square. So, that means it can't be simplified into an integer.

But maybe it can be broken down into a product of a square number and another number? Let me try factoring 53. Wait, 53 is a prime number, right? Let me check. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Let me test divisibility.

Is 53 divisible by 2? No, it's odd. Divisible by 3? Let's see, 5 + 3 = 8, and 8 isn't divisible by 3, so no. Divisible by 5? It doesn't end with 0 or 5, so no. Next prime is 7. 7 times 7 is 49, 7 times 8 is 56, which is more than 53. So 53 isn't divisible by 7. The next prime is 11, but 11 times 5 is 55, which is more than 53. So yeah, 53 is a prime number.

Since 53 is prime, its only factors are 1 and 53. Therefore, when we try to break down √53, we can't factor it into any numbers other than 1 and 53. Since 1 is a perfect square (1 squared is 1), but that doesn't help us simplify the radical. So, there are no perfect square factors other than 1.

Therefore, √53 cannot be simplified further. So, the simplest radical form of √53 is just √53.

Wait, but let me double-check. Maybe I made a mistake in assuming that 53 is prime? Let me try dividing 53 by some other numbers. For example, 53 divided by 13 is about 4.07, which isn't an integer. 17? 17 times 3 is 51, 17 times 4 is 68. Nope. 19? 19 times 2 is 38, 19 times 3 is 57. Nope. So, yeah, 53 is definitely a prime number.

Therefore, there are no square factors other than 1. So, √53 is already in its simplest radical form.

But just to make sure, let me recall the definition of simplest radical form again. It requires that the radicand has no perfect square factors other than 1, and that the index of the radical is as small as possible. Since the index here is 2 (square root), and there are no perfect square factors, it's already simplified.

Alternatively, if there was a factor like 4, 9, 16, etc., we could take that out of the square root. For example, √18 can be simplified to 3√2 because 18 = 9*2, and 9 is a perfect square. But in this case, since 53 is prime, there's no such factor.

Another way to think about it is that if you try to factor 53 into two numbers, say a and b, such that a*b = 53, then one of them has to be 1 and the other 53. Since 1 is a perfect square, but that doesn't help. So, there's no way to write 53 as a product of a perfect square and another number greater than 1.

Therefore, I think I can confidently say that √53 is already in its simplest radical form.

But just to be thorough, let me check if there's any other way to represent it. For example, sometimes people might think about rationalizing the denominator or something, but that's when you have a radical in the denominator. In this case, there's no denominator, so that doesn't apply.

Alternatively, could it be expressed as a sum or difference of square roots? Like, maybe √53 can be written as √a - √b or something? But that's not really simplifying it; that's more of an expansion. And I don't think that's necessary here.

Also, if I approximate √53, it's about 7.28, but that's a decimal approximation, not a radical form. The question specifically asks for simplest radical form, so decimal isn't the answer here.

So, after all that, I think the answer is just √53.

**Final Answer**
The simplest radical form of $\sqrt{53}$ is \boxed{\sqrt{53}}.

To determine the simplest radical form of $\sqrt{53}$, we begin by understanding what "simplest radical form" entails. A square root is in simplest radical form if:

1. The number under the square root (the **radicand**) has **no perfect square factors** other than 1.
2. The **index of the radical** (in this case, 2 for square roots) is as small as possible.

---

### Step 1: Check if 53 is a perfect square

We know that:
- $7^2 = 49$
- $8^2 = 64$

Since $53$ lies between $49$ and $64$, it is **not a perfect square**.

---

### Step 2: Factor 53 to check for perfect square factors

We now examine the prime factorization of 53. Since 53 is a **prime number**, its only factors are 1 and 53. This means:

- The only perfect square factor of 53 is $1$, which doesn't help in simplifying the square root.

---

### Step 3: Conclusion

Because 53 has no perfect square factors other than 1, the square root $\sqrt{53}$ **cannot be simplified further**. Therefore, it is already in its **simplest radical form**.

---

### Final Answer

$$
\boxed{\sqrt{53}}
$$

Solution: Since 53 is prime, $\boxed{\sqrt{53}}$ is already in simplest radical form.