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 some basics 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, translating that into an equation, it would be:

Interior angle = 6.5 * Exterior angle

Which, using the variables I just mentioned, would be:

180 - x = 6.5x

Wait, let me check that again. If the interior angle is 6.5 times the exterior angle, then yes, that equation makes sense. Let me solve for x here.

So, 180 - x = 6.5x

Adding x to both sides:

180 = 7.5x

Then, dividing both sides by 7.5:

x = 180 / 7.5

Hmm, let me calculate that. 7.5 times 24 is 180, right? Because 7.5 * 20 = 150, and 7.5 * 4 = 30, so 150 + 30 = 180. So, x = 24 degrees.

So, the measure of each exterior angle is 24 degrees. Now, I need to find the number of sides of the polygon.

I remember that the sum of all exterior angles of any polygon is 360 degrees. And since it's a regular polygon, all exterior angles are equal. Therefore, the measure of each exterior angle is 360 divided by the number of sides, n.

So, if each exterior angle is 24 degrees, then:

24 = 360 / n

Solving for n:

n = 360 / 24

Calculating that, 360 divided by 24. Let me do this step by step. 24 times 10 is 240, subtract that from 360, you get 120. 24 times 5 is 120. So, 10 + 5 = 15. Therefore, n = 15.

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

If it's a regular 15-gon, each exterior angle is 360/15 = 24 degrees, which matches our earlier calculation. Then the interior angle would be 180 - 24 = 156 degrees. Let me check if 156 is indeed 6.5 times 24.

Calculating 6.5 * 24. Well, 6 * 24 = 144, and 0.5 * 24 = 12, so 144 + 12 = 156. Yes, that's correct.

So, the number of sides is 15.

But let me just make sure I didn't make any mistakes in my reasoning. Let me go through the steps again.

1. Let exterior angle = x. Then interior angle = 180 - x.

2. Given that interior angle = 6.5 * exterior angle, so 180 - x = 6.5x.

3. Solving for x: 180 = 7.5x => x = 24.

4. Then, since exterior angle is 360/n, so n = 360 / 24 = 15.

All steps seem logical. Also, checking the final numbers confirm that the interior angle is indeed 6.5 times the exterior angle.

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

I know that the measure of each interior angle of a regular polygon is given by:

Interior angle = [(n - 2) * 180] / n

And the measure of each exterior angle is:

Exterior angle = 360 / n

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

[(n - 2) * 180] / n = 6.5 * (360 / n)

Hmm, let me write that equation:

[(n - 2) * 180] / n = 6.5 * (360 / n)

Since both sides have a denominator of n, I can multiply both sides by n to eliminate denominators:

(n - 2) * 180 = 6.5 * 360

Calculating the right side: 6.5 * 360. Let me compute that.

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

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

Divide both sides by 180:

n - 2 = 2340 / 180

Calculating that division. Let me simplify.

Divide numerator and denominator by 10: 234 / 18.

Divide numerator and denominator by 9: 26 / 2 = 13. Wait, 234 divided by 18: 18*13 = 234. Yes, because 18*10=180, 18*3=54, so 180+54=234. Therefore, 2340 / 180 = 13.

Therefore, n - 2 = 13 => 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, and checking the numbers works out. So, I'm confident that the answer is 15.

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

To determine the number of sides of 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: Define Variables

Let the measure of an **exterior angle** be $ x $ degrees.
Then, the measure of the corresponding **interior angle** is $ 180^\circ - x $, since interior and exterior angles are supplementary.

We are given that:

$$
\text{Interior angle} = 6.5 \times \text{Exterior angle}
$$

Substituting the expressions:

$$
180 - x = 6.5x
$$

---

### Step 2: Solve for $ x $

$$
180 = 6.5x + x = 7.5x
$$

$$
x = \frac{180}{7.5} = 24
$$

So, the measure of each **exterior angle** is $ 24^\circ $.

---

### Step 3: Use the Exterior Angle Sum Theorem

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

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

---

### Step 4: Verification

- Each **exterior angle** = $ 24^\circ $
- Each **interior angle** = $ 180 - 24 = 156^\circ $
- Check if $ 156 = 6.5 \times 24 $:
  $ 6.5 \times 24 = 156 $, which is correct.

---

### 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 some basics 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, translating that into an equation, it would be:

Interior angle = 6.5 * Exterior angle

Which, using the variables I just mentioned, would be:

180 - x = 6.5x

Wait, let me check that again. If the interior angle is 6.5 times the exterior angle, then yes, that equation makes sense. Let me solve for x here.

So, 180 - x = 6.5x

Adding x to both sides:

180 = 7.5x

Then, dividing both sides by 7.5:

x = 180 / 7.5

Hmm, let me calculate that. 7.5 times 24 is 180, right? Because 7.5 * 20 = 150, and 7.5 * 4 = 30, so 150 + 30 = 180. So, x = 24 degrees.

So, the measure of each exterior angle is 24 degrees. Now, I need to find the number of sides of the polygon.

I remember that the sum of all exterior angles of any polygon is 360 degrees. And since it's a regular polygon, all exterior angles are equal. Therefore, the measure of each exterior angle is 360 divided by the number of sides, n.

So, if each exterior angle is 24 degrees, then:

24 = 360 / n

Solving for n:

n = 360 / 24

Calculating that, 360 divided by 24. Let me do this step by step. 24 times 10 is 240, subtract that from 360, you get 120. 24 times 5 is 120. So, 10 + 5 = 15. Therefore, n = 15.

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

If it's a regular 15-gon, each exterior angle is 360/15 = 24 degrees, which matches our earlier calculation. Then the interior angle would be 180 - 24 = 156 degrees. Let me check if 156 is indeed 6.5 times 24.

Calculating 6.5 * 24. Well, 6 * 24 = 144, and 0.5 * 24 = 12, so 144 + 12 = 156. Yes, that's correct.

So, the number of sides is 15.

But let me just make sure I didn't make any mistakes in my reasoning. Let me go through the steps again.

1. Let exterior angle = x. Then interior angle = 180 - x.

2. Given that interior angle = 6.5 * exterior angle, so 180 - x = 6.5x.

3. Solving for x: 180 = 7.5x => x = 24.

4. Then, since exterior angle is 360/n, so n = 360 / 24 = 15.

All steps seem logical. Also, checking the final numbers confirm that the interior angle is indeed 6.5 times the exterior angle.

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

I know that the measure of each interior angle of a regular polygon is given by:

Interior angle = [(n - 2) * 180] / n

And the measure of each exterior angle is:

Exterior angle = 360 / n

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

[(n - 2) * 180] / n = 6.5 * (360 / n)

Hmm, let me write that equation:

[(n - 2) * 180] / n = 6.5 * (360 / n)

Since both sides have a denominator of n, I can multiply both sides by n to eliminate denominators:

(n - 2) * 180 = 6.5 * 360

Calculating the right side: 6.5 * 360. Let me compute that.

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

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

Divide both sides by 180:

n - 2 = 2340 / 180

Calculating that division. Let me simplify.

Divide numerator and denominator by 10: 234 / 18.

Divide numerator and denominator by 9: 26 / 2 = 13. Wait, 234 divided by 18: 18*13 = 234. Yes, because 18*10=180, 18*3=54, so 180+54=234. Therefore, 2340 / 180 = 13.

Therefore, n - 2 = 13 => 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, and checking the numbers works out. So, I'm confident that the answer is 15.

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

To determine the number of sides of 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: Define Variables

Let the measure of an **exterior angle** be $ x $ degrees.
Then, the measure of the corresponding **interior angle** is $ 180^\circ - x $, since interior and exterior angles are supplementary.

We are given that:

$$
\text{Interior angle} = 6.5 \times \text{Exterior angle}
$$

Substituting the expressions:

$$
180 - x = 6.5x
$$

---

### Step 2: Solve for $ x $

$$
180 = 6.5x + x = 7.5x
$$

$$
x = \frac{180}{7.5} = 24
$$

So, the measure of each **exterior angle** is $ 24^\circ $.

---

### Step 3: Use the Exterior Angle Sum Theorem

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

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

---

### Step 4: Verification

- Each **exterior angle** = $ 24^\circ $
- Each **interior angle** = $ 180 - 24 = 156^\circ $
- Check if $ 156 = 6.5 \times 24 $:
  $ 6.5 \times 24 = 156 $, which is correct.

---

### 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:

1331 = 11^3

9999 = 9 * 11 * 101 = 9 * 11 * 101

So, the GCD of 1331 and 9999 is the product of the common prime factors with the lowest exponents. Both have 11 as a common factor. 1331 has 11^3, and 9999 has 11^1. So the GCD is 11^1 = 11.

Therefore, we can divide numerator and denominator by 11:

1331 ÷ 11 = 121

9999 ÷ 11 = 909

Wait, let me check that division. 11*121 = 1331, which is correct. And 9999 ÷ 11. Let me compute that. 11*900 = 9900, so 9999 - 9900 = 99. 99 ÷ 11 = 9. So total is 900 + 9 = 909. Yes, so 9999 ÷ 11 = 909.

So now the fraction is 121/909. Wait, but is this reduced completely? Let me check if 121 and 909 have any common factors.

First, factor 121. 121 is 11 squared. So 11^2.

Now, 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*3*101 = 3^2 * 101.

So, 121 is 11^2, and 909 is 3^2 * 101. There are no common prime factors between numerator and denominator. 11 is not a factor of 909, since 909 ÷ 11 is... Let me check. 11*82 = 902, 909 - 902 = 7, so no. So 11 doesn't divide into 909. Similarly, 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 = 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 verify the steps again.

Original decimal: 0.133113311331...

Set x = 0.13311331...

Multiply by 10^4 = 10000: 10000x = 1331.13311331...

Subtract x: 10000x - x = 1331.1331... - 0.1331... => 9999x = 1331 => x = 1331/9999.

Then GCD(1331, 9999). As above, 1331 is 11^3, 9999 is 9*11*101. So GCD is 11. Dividing numerator and denominator by 11 gives 121/909.

Then 121 is 11^2, 909 is 3^2*101. No common factors. So yes, 121/909 is reduced. Therefore, a + b is 121 + 909 = 1030.

Wait, but let me check if 121 and 909 have any common factors. Let me check with Euclidean algorithm.

Compute GCD(121, 909).

First, divide 909 by 121. 121*7 = 847. 909 - 847 = 62. So GCD(121, 62).

Now divide 121 by 62: 62*1 = 62, remainder 59. GCD(62, 59).

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

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

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

Divide 2 by 1: remainder 0. So GCD is 1.

Therefore, yes, 121 and 909 are coprime. Therefore, the fraction is reduced completely, so a + b is 121 + 909 = 1030.

But wait, let me check once more. Maybe I made a mistake in the initial step? Let me verify with another approach.

Alternatively, maybe I can write the decimal as 0.13311331... which is 0.1331 repeating. Let me think if there's another way to represent this.

Alternatively, maybe I can think of the decimal as 0.1331 + 0.00001331 + 0.000000001331 + ... which is a geometric series.

But that might be more complicated. Let me try.

The decimal is 0.133113311331... So the first four digits after the decimal are 1331, then it repeats. So the decimal can be considered as the sum from n=0 to infinity of (1331 * 10^{-4(n+1)}). Wait, maybe?

Wait, the first term is 1331 * 10^{-4}, then the next term is 1331 * 10^{-8}, then 1331 * 10^{-12}, etc. So it's a geometric series with first term a = 1331 * 10^{-4} and common ratio r = 10^{-4}.

The sum of an infinite geometric series is a / (1 - r). Therefore, the total is (1331 * 10^{-4}) / (1 - 10^{-4}) = (1331 / 10^4) / ( (10^4 - 1)/10^4 ) ) = 1331 / (10^4 - 1) = 1331 / 9999. Which is the same result as before. So that confirms that x = 1331 / 9999.

Therefore, my initial calculation is correct. Then simplifying gives 121/909, and a + b = 1030.

But just to be thorough, let me check if 121/909 is indeed equal to 0.13311331...

Let me perform the division 121 ÷ 909.

Alternatively, since 121/909 is the simplified fraction, let me check if multiplying numerator and denominator by something gives me the original decimal.

Alternatively, let me check 121 divided by 909. Let me compute 121 ÷ 909.

Well, 909 goes into 121 zero times. Add decimal: 121.000...

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 where we started after the first division step. So the decimal repeats here.

So the decimal expansion is 0.1331331..., which is 0.13311331... Wait, hold on. Wait, let me write down the decimal digits:

First step: 0.

After first division: 0.1 (since 1), remainder 301.

Second step: 3, so 0.13, remainder 283.

Third step: 3, so 0.133, remainder 103.

Fourth step: 1, so 0.1331, remainder 121.

Then we bring down a zero, making 1210, which is the same as the first step. So the decimal repeats from here.

Therefore, the decimal expansion is 0.1331 1331 1331..., which is 0.133113311331..., 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.

But just to be 100% sure, let me check if there's a different way someone might approach this problem and maybe get a different answer. For example, sometimes when there's a repeating decimal, the number of repeating digits can affect the multiplier. Let me confirm that the repeating part is indeed four digits.

The decimal is 0.133113311331..., so the repeating part is "1331", which is four digits. So the standard method applies. If the repeating part was shorter, like if it was 0.1331331331..., then the repeating part would be "331", but in this case, the bar is over 1331, so the entire four digits repeat.

Alternatively, if someone thought the repeating part was only three digits, they might have made a mistake, but since the problem says 0.\overline{1331}, that notation means that the bar is over the four digits 1331, so it's definitely four digits.

Another check: Let me compute 121 divided by 909. Let me use another method. Let me note that 121/909. Let me multiply numerator and denominator by something to make the denominator 10000 - 1? Wait, 909 is 9*101. Wait, but maybe not helpful.

Alternatively, since we know that 1331/9999 reduces to 121/909, and we confirmed that 121/909 is equal to the decimal, then that's correct.

Alternatively, maybe I can check with another fraction. Let me compute 121/909. Let me approximate it as a decimal.

Well, 121 divided by 909. Let me approximate. Since 121/909 ≈ 121/900 ≈ 0.1344... But the actual decimal is 0.13311331..., which is approximately 0.1331. So 0.1331 is less than 0.1344. Let me compute 121/909 more accurately.

Let me do the division step by step again:

Divide 121 by 909:

0.
909 goes into 1210 once (1), 1*909=909, subtract from 1210: 301. Bring down 0: 3010.

909 goes into 3010 three times (3), 3*909=2727, subtract from 3010: 283. Bring down 0: 2830.

909 goes into 2830 three times (3), 3*909=2727, subtract: 2830 - 2727 = 103. Bring down 0: 1030.

909 goes into 1030 once (1), 1*909=909, subtract: 1030 - 909 = 121. Bring down 0: 1210.

Now we are back to where we started after the first step. So the decimal is 0.1331 1331..., so 0.13311331..., which matches the original decimal. Therefore, the fraction is correct.

Therefore, I think my answer is correct. The fraction is 121/909, so a + b is 121 + 909 = 1030.

But just to be absolutely thorough, let me check if there's a possibility that the original fraction could be simplified further. For example, maybe I made a mistake in calculating the GCD? Let me check GCD(1331, 9999) again.

As before, 1331 is 11^3, and 9999 is 9*11*101. Therefore, GCD is 11. So dividing numerator and denominator by 11 gives 121/909.

Alternatively, if I use the Euclidean algorithm on 1331 and 9999:

GCD(9999, 1331).

Compute 9999 divided by 1331.

1331*7 = 9317. 9999 - 9317 = 682. So GCD(1331, 682).

Now compute GCD(1331, 682).

1331 divided by 682 is 1 with remainder 1331 - 682 = 649. So GCD(682, 649).

682 divided by 649 is 1 with remainder 33. GCD(649, 33).

649 divided by 33 is 19*33 = 627, remainder 649 - 627 = 22. GCD(33, 22).

33 divided by 22 is 1 with remainder 11. GCD(22, 11).

22 divided by 11 is 2 with remainder 0. So GCD is 11.

Therefore, GCD(1331, 9999) is indeed 11, so dividing by 11 gives 121/909.

Therefore, the fraction is reduced completely.

Therefore, the answer is 121 + 909 = 1030.

I think that's solid. I can't find any mistakes in my reasoning. So I think 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 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:

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

$$
9999x = 1331
$$

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

---

### 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:

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

---

### Step 4: Confirm the fraction is in lowest terms

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 5: Compute $ a + b $

The 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:

1331 = 11^3

9999 = 9 * 11 * 101 = 9 * 11 * 101

So, the GCD of 1331 and 9999 is the product of the common prime factors with the lowest exponents. Both have 11 as a common factor. 1331 has 11^3, and 9999 has 11^1. So the GCD is 11^1 = 11.

Therefore, we can divide numerator and denominator by 11:

1331 ÷ 11 = 121

9999 ÷ 11 = 909

Wait, let me check that division. 11*121 = 1331, which is correct. And 9999 ÷ 11. Let me compute that. 11*900 = 9900, so 9999 - 9900 = 99. 99 ÷ 11 = 9. So total is 900 + 9 = 909. Yes, so 9999 ÷ 11 = 909.

So now the fraction is 121/909. Wait, but is this reduced completely? Let me check if 121 and 909 have any common factors.

First, factor 121. 121 is 11 squared. So 11^2.

Now, 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*3*101 = 3^2 * 101.

So, 121 is 11^2, and 909 is 3^2 * 101. There are no common prime factors between numerator and denominator. 11 is not a factor of 909, since 909 ÷ 11 is... Let me check. 11*82 = 902, 909 - 902 = 7, so no. So 11 doesn't divide into 909. Similarly, 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 = 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 verify the steps again.

Original decimal: 0.133113311331...

Set x = 0.13311331...

Multiply by 10^4 = 10000: 10000x = 1331.13311331...

Subtract x: 10000x - x = 1331.1331... - 0.1331... => 9999x = 1331 => x = 1331/9999.

Then GCD(1331, 9999). As above, 1331 is 11^3, 9999 is 9*11*101. So GCD is 11. Dividing numerator and denominator by 11 gives 121/909.

Then 121 is 11^2, 909 is 3^2*101. No common factors. So yes, 121/909 is reduced. Therefore, a + b is 121 + 909 = 1030.

Wait, but let me check if 121 and 909 have any common factors. Let me check with Euclidean algorithm.

Compute GCD(121, 909).

First, divide 909 by 121. 121*7 = 847. 909 - 847 = 62. So GCD(121, 62).

Now divide 121 by 62: 62*1 = 62, remainder 59. GCD(62, 59).

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

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

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

Divide 2 by 1: remainder 0. So GCD is 1.

Therefore, yes, 121 and 909 are coprime. Therefore, the fraction is reduced completely, so a + b is 121 + 909 = 1030.

But wait, let me check once more. Maybe I made a mistake in the initial step? Let me verify with another approach.

Alternatively, maybe I can write the decimal as 0.13311331... which is 0.1331 repeating. Let me think if there's another way to represent this.

Alternatively, maybe I can think of the decimal as 0.1331 + 0.00001331 + 0.000000001331 + ... which is a geometric series.

But that might be more complicated. Let me try.

The decimal is 0.133113311331... So the first four digits after the decimal are 1331, then it repeats. So the decimal can be considered as the sum from n=0 to infinity of (1331 * 10^{-4(n+1)}). Wait, maybe?

Wait, the first term is 1331 * 10^{-4}, then the next term is 1331 * 10^{-8}, then 1331 * 10^{-12}, etc. So it's a geometric series with first term a = 1331 * 10^{-4} and common ratio r = 10^{-4}.

The sum of an infinite geometric series is a / (1 - r). Therefore, the total is (1331 * 10^{-4}) / (1 - 10^{-4}) = (1331 / 10^4) / ( (10^4 - 1)/10^4 ) ) = 1331 / (10^4 - 1) = 1331 / 9999. Which is the same result as before. So that confirms that x = 1331 / 9999.

Therefore, my initial calculation is correct. Then simplifying gives 121/909, and a + b = 1030.

But just to be thorough, let me check if 121/909 is indeed equal to 0.13311331...

Let me perform the division 121 ÷ 909.

Alternatively, since 121/909 is the simplified fraction, let me check if multiplying numerator and denominator by something gives me the original decimal.

Alternatively, let me check 121 divided by 909. Let me compute 121 ÷ 909.

Well, 909 goes into 121 zero times. Add decimal: 121.000...

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 where we started after the first division step. So the decimal repeats here.

So the decimal expansion is 0.1331331..., which is 0.13311331... Wait, hold on. Wait, let me write down the decimal digits:

First step: 0.

After first division: 0.1 (since 1), remainder 301.

Second step: 3, so 0.13, remainder 283.

Third step: 3, so 0.133, remainder 103.

Fourth step: 1, so 0.1331, remainder 121.

Then we bring down a zero, making 1210, which is the same as the first step. So the decimal repeats from here.

Therefore, the decimal expansion is 0.1331 1331 1331..., which is 0.133113311331..., 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.

But just to be 100% sure, let me check if there's a different way someone might approach this problem and maybe get a different answer. For example, sometimes when there's a repeating decimal, the number of repeating digits can affect the multiplier. Let me confirm that the repeating part is indeed four digits.

The decimal is 0.133113311331..., so the repeating part is "1331", which is four digits. So the standard method applies. If the repeating part was shorter, like if it was 0.1331331331..., then the repeating part would be "331", but in this case, the bar is over 1331, so the entire four digits repeat.

Alternatively, if someone thought the repeating part was only three digits, they might have made a mistake, but since the problem says 0.\overline{1331}, that notation means that the bar is over the four digits 1331, so it's definitely four digits.

Another check: Let me compute 121 divided by 909. Let me use another method. Let me note that 121/909. Let me multiply numerator and denominator by something to make the denominator 10000 - 1? Wait, 909 is 9*101. Wait, but maybe not helpful.

Alternatively, since we know that 1331/9999 reduces to 121/909, and we confirmed that 121/909 is equal to the decimal, then that's correct.

Alternatively, maybe I can check with another fraction. Let me compute 121/909. Let me approximate it as a decimal.

Well, 121 divided by 909. Let me approximate. Since 121/909 ≈ 121/900 ≈ 0.1344... But the actual decimal is 0.13311331..., which is approximately 0.1331. So 0.1331 is less than 0.1344. Let me compute 121/909 more accurately.

Let me do the division step by step again:

Divide 121 by 909:

0.
909 goes into 1210 once (1), 1*909=909, subtract from 1210: 301. Bring down 0: 3010.

909 goes into 3010 three times (3), 3*909=2727, subtract from 3010: 283. Bring down 0: 2830.

909 goes into 2830 three times (3), 3*909=2727, subtract: 2830 - 2727 = 103. Bring down 0: 1030.

909 goes into 1030 once (1), 1*909=909, subtract: 1030 - 909 = 121. Bring down 0: 1210.

Now we are back to where we started after the first step. So the decimal is 0.1331 1331..., so 0.13311331..., which matches the original decimal. Therefore, the fraction is correct.

Therefore, I think my answer is correct. The fraction is 121/909, so a + b is 121 + 909 = 1030.

But just to be absolutely thorough, let me check if there's a possibility that the original fraction could be simplified further. For example, maybe I made a mistake in calculating the GCD? Let me check GCD(1331, 9999) again.

As before, 1331 is 11^3, and 9999 is 9*11*101. Therefore, GCD is 11. So dividing numerator and denominator by 11 gives 121/909.

Alternatively, if I use the Euclidean algorithm on 1331 and 9999:

GCD(9999, 1331).

Compute 9999 divided by 1331.

1331*7 = 9317. 9999 - 9317 = 682. So GCD(1331, 682).

Now compute GCD(1331, 682).

1331 divided by 682 is 1 with remainder 1331 - 682 = 649. So GCD(682, 649).

682 divided by 649 is 1 with remainder 33. GCD(649, 33).

649 divided by 33 is 19*33 = 627, remainder 649 - 627 = 22. GCD(33, 22).

33 divided by 22 is 1 with remainder 11. GCD(22, 11).

22 divided by 11 is 2 with remainder 0. So GCD is 11.

Therefore, GCD(1331, 9999) is indeed 11, so dividing by 11 gives 121/909.

Therefore, the fraction is reduced completely.

Therefore, the answer is 121 + 909 = 1030.

I think that's solid. I can't find any mistakes in my reasoning. So I think 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 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:

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

$$
9999x = 1331
$$

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

---

### 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:

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

---

### Step 4: Confirm the fraction is in lowest terms

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 5: Compute $ a + b $

The 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 calculate that.

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

Which is approximately 833.333...

But let me do exact division.

10,000 divided by 12.

Well, 12 * 833 = 9996.

So, 10,000 - 9996 = 4.

So, 10,000 / 12 = 833 + 4/12 = 833 + 1/3 ≈ 833.333...

So, then multiply that by $1.50.

So, 833.333... * 1.50.

Let me compute that.

First, note that 833.333... is equal to 2500/3. Because 833.333... is 2500 divided by 3. Let me check: 2500 / 3 ≈ 833.333...

Yes.

So, 2500/3 * 1.5.

But 1.5 is 3/2.

So, 2500/3 * 3/2 = 2500/2 = 1250.

So, again, same answer.

Therefore, regardless of the method, I get $1,250.

So, I think that's solid.

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 that rate by 1,000,000 lire to get the dollar amount: 1,000,000 * 0.00125 = 1,250.

Alternatively, using proportions or unit conversion, all methods lead to the same answer.

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

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

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 1 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 the number of lire by the exchange rate:

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

---

### 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:

$$
1200x = 1,000,000 \times 1.50
$$

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

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 calculate that.

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

Which is approximately 833.333...

But let me do exact division.

10,000 divided by 12.

Well, 12 * 833 = 9996.

So, 10,000 - 9996 = 4.

So, 10,000 / 12 = 833 + 4/12 = 833 + 1/3 ≈ 833.333...

So, then multiply that by $1.50.

So, 833.333... * 1.50.

Let me compute that.

First, note that 833.333... is equal to 2500/3. Because 833.333... is 2500 divided by 3. Let me check: 2500 / 3 ≈ 833.333...

Yes.

So, 2500/3 * 1.5.

But 1.5 is 3/2.

So, 2500/3 * 3/2 = 2500/2 = 1250.

So, again, same answer.

Therefore, regardless of the method, I get $1,250.

So, I think that's solid.

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 that rate by 1,000,000 lire to get the dollar amount: 1,000,000 * 0.00125 = 1,250.

Alternatively, using proportions or unit conversion, all methods lead to the same answer.

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

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

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 1 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 the number of lire by the exchange rate:

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

---

### 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:

$$
1200x = 1,000,000 \times 1.50
$$

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

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: factor 53. Let me think. 53 is a prime number, right? Wait, is it? Let me check. Let me try dividing 53 by some primes. 2? No, it's odd. 3? 3 times 17 is 51, 3 times 18 is 54, so no. 5? Ends with 5 or 0, so no. 7? 7 times 7 is 49, 7 times 8 is 56, so no. 11? 11 times 4 is 44, 11 times 5 is 55. So no. So 53 is a prime number. Therefore, its only factors are 1 and 53.

So, since 53 is prime, it doesn't have any perfect square factors other than 1. Therefore, the square root of 53 cannot be simplified further. So, is that it? Wait, but let me make sure I didn't miss anything. Maybe I should check if there's a way to express it as a product of a square and another number. For example, if I have something like sqrt(16*3) = 4*sqrt(3). But in this case, since 53 is prime, there's no such factorization.

Alternatively, maybe I can check if 53 is a square number? Well, 7 squared is 49, 8 squared is 64. So 53 is between those two, so it's not a perfect square. Therefore, sqrt(53) is irrational and cannot be simplified.

Wait, but maybe I need to check if there's a mistake in my thinking. Let me think again. Suppose someone says that 53 can be factored into something else? Let me try dividing 53 by primes again. 2, 3, 5, 7, 11... up to sqrt(53) which is approximately 7.28. So primes up to 7. So I already checked up to 7. So 53 is prime. Therefore, there are no factors other than 1 and itself.

Therefore, sqrt(53) is already in its simplest radical form. So the answer is just sqrt(53).

But let me check with another approach. Let me try to see if there's any square number that divides 53. The square numbers less than 53 are 1, 4, 9, 16, 25, 36, 49. Let me check if any of these divide 53.

53 divided by 1 is 53, which is an integer. 53 divided by 4 is 13.25, not integer. Divided by 9 is about 5.88, nope. Divided by 16 is 3.3125, nope. Divided by 25 is 2.12, nope. Divided by 36 is 1.472..., nope. Divided by 49 is about 1.08... nope. So none of the square numbers except 1 divide 53. Therefore, sqrt(53) cannot be simplified.

Therefore, the simplest radical form is just sqrt(53).

But wait, maybe I need to write it as something else? Like, maybe combining with other radicals? But no, since there's no other factors.

Alternatively, maybe the question is trying to trick me? Let me check if 53 is a multiple of any square numbers. For example, if it was 50, then sqrt(50) is 5*sqrt(2). But 53 is prime.

Alternatively, maybe I made a mistake in determining if 53 is prime? Let me check again. Let me try dividing 53 by 7. 7*7 is 49, 7*8 is 56. So 53-49 is 4, so remainder 4. Not divisible by 7. Divided by 3? 3*17 is 51, 51+3=54, so remainder 2. Not divisible by 3. Divided by 5? Ends with 3, so no. Divided by 2? No. So yes, 53 is prime.

Therefore, sqrt(53) is already in simplest radical form.

Wait, but maybe the question is expecting me to write it as a decimal? But no, simplest radical form refers to the expression with the radical, not a decimal approximation. So, since it can't be simplified, the answer is sqrt(53).

Alternatively, maybe there's a different way to represent it? For example, sometimes people might think of breaking it into sum of squares or something, but that's not relevant for simplifying radicals.

Alternatively, maybe I need to check if 53 is a square of a fraction? But that would not make sense here.

Alternatively, maybe I need to check if there's a mistake in the problem? Like, maybe it was supposed to be sqrt(50) or something else? But the problem says sqrt(53).

Alternatively, maybe I need to check if 53 is a perfect square in some other way? But no, as we saw, it's between 7^2 and 8^2.

Therefore, I think my conclusion is correct. The square root of 53 is already in its simplest radical form.

But just to make sure, let me check an example. For example, sqrt(12) can be simplified to 2*sqrt(3) because 12=4*3. But since 53 has no such factors, it can't be simplified.

Another example: sqrt(18) is 3*sqrt(2). Similarly, sqrt(50) is 5*sqrt(2). But sqrt(53) can't be broken down.

Therefore, I think the answer is sqrt(53).

Wait, but maybe the question is expecting me to write it as something else? Let me check with another method. Let me try prime factorization of 53. Since 53 is prime, its prime factorization is just 53. Therefore, sqrt(53) is sqrt(53), which can't be simplified.

Alternatively, if I write it as sqrt(53) = sqrt(1*53) = sqrt(1)*sqrt(53) = 1*sqrt(53) = sqrt(53). So that's the same thing.

Therefore, there's no way to simplify it further.

Therefore, I think the answer is sqrt(53).

But just to be thorough, let me check an online source or something. Wait, I can't actually access external resources, but I can recall that 53 is indeed a prime number. For example, primes between 1-100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, etc. So yes, 53 is a prime number.

Therefore, sqrt(53) cannot be simplified.

So, the simplest radical form is just sqrt(53).

Therefore, the answer is \boxed{\sqrt{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" means. A square root is in simplest radical form when the number under the square root (the radicand) has no perfect square factors other than 1. This means we should check whether 53 can be factored into a product that includes a perfect square.

---

### Step 1: Prime Factorization of 53

We start by factoring 53. Since 53 is an odd number, it is not divisible by 2. It is not divisible by 3 (since $5 + 3 = 8$, which is not divisible by 3), nor by 5 (it doesn't end in 0 or 5). Testing divisibility by 7, we find that $7 \times 7 = 49$ and $7 \times 8 = 56$, so 53 is not divisible by 7. Continuing this process, we find that 53 is not divisible by any prime number less than or equal to $\sqrt{53} \approx 7.28$. Therefore, **53 is a prime number**.

---

### Step 2: Check for Perfect Square Factors

Since 53 is prime, its only factors are 1 and itself. This means that the only perfect square factor of 53 is 1. Therefore, $\sqrt{53}$ cannot be simplified further by factoring out any perfect square.

---

### Step 3: Confirm That 53 Is Not a Perfect Square

To confirm, we note that $7^2 = 49$ and $8^2 = 64$, so 53 lies between these two perfect squares. Since it is not a perfect square, $\sqrt{53}$ is irrational and cannot be expressed as a simplified radical with a smaller radicand.

---

### Final Conclusion

Because 53 is a prime number and has no perfect square factors other than 1, the square root of 53 is already in its simplest radical form.

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

[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: factor 53. Let me think. 53 is a prime number, right? Wait, is it? Let me check. Let me try dividing 53 by some primes. 2? No, it's odd. 3? 3 times 17 is 51, 3 times 18 is 54, so no. 5? Ends with 5 or 0, so no. 7? 7 times 7 is 49, 7 times 8 is 56, so no. 11? 11 times 4 is 44, 11 times 5 is 55. So no. So 53 is a prime number. Therefore, its only factors are 1 and 53.

So, since 53 is prime, it doesn't have any perfect square factors other than 1. Therefore, the square root of 53 cannot be simplified further. So, is that it? Wait, but let me make sure I didn't miss anything. Maybe I should check if there's a way to express it as a product of a square and another number. For example, if I have something like sqrt(16*3) = 4*sqrt(3). But in this case, since 53 is prime, there's no such factorization.

Alternatively, maybe I can check if 53 is a square number? Well, 7 squared is 49, 8 squared is 64. So 53 is between those two, so it's not a perfect square. Therefore, sqrt(53) is irrational and cannot be simplified.

Wait, but maybe I need to check if there's a mistake in my thinking. Let me think again. Suppose someone says that 53 can be factored into something else? Let me try dividing 53 by primes again. 2, 3, 5, 7, 11... up to sqrt(53) which is approximately 7.28. So primes up to 7. So I already checked up to 7. So 53 is prime. Therefore, there are no factors other than 1 and itself.

Therefore, sqrt(53) is already in its simplest radical form. So the answer is just sqrt(53).

But let me check with another approach. Let me try to see if there's any square number that divides 53. The square numbers less than 53 are 1, 4, 9, 16, 25, 36, 49. Let me check if any of these divide 53.

53 divided by 1 is 53, which is an integer. 53 divided by 4 is 13.25, not integer. Divided by 9 is about 5.88, nope. Divided by 16 is 3.3125, nope. Divided by 25 is 2.12, nope. Divided by 36 is 1.472..., nope. Divided by 49 is about 1.08... nope. So none of the square numbers except 1 divide 53. Therefore, sqrt(53) cannot be simplified.

Therefore, the simplest radical form is just sqrt(53).

But wait, maybe I need to write it as something else? Like, maybe combining with other radicals? But no, since there's no other factors.

Alternatively, maybe the question is trying to trick me? Let me check if 53 is a multiple of any square numbers. For example, if it was 50, then sqrt(50) is 5*sqrt(2). But 53 is prime.

Alternatively, maybe I made a mistake in determining if 53 is prime? Let me check again. Let me try dividing 53 by 7. 7*7 is 49, 7*8 is 56. So 53-49 is 4, so remainder 4. Not divisible by 7. Divided by 3? 3*17 is 51, 51+3=54, so remainder 2. Not divisible by 3. Divided by 5? Ends with 3, so no. Divided by 2? No. So yes, 53 is prime.

Therefore, sqrt(53) is already in simplest radical form.

Wait, but maybe the question is expecting me to write it as a decimal? But no, simplest radical form refers to the expression with the radical, not a decimal approximation. So, since it can't be simplified, the answer is sqrt(53).

Alternatively, maybe there's a different way to represent it? For example, sometimes people might think of breaking it into sum of squares or something, but that's not relevant for simplifying radicals.

Alternatively, maybe I need to check if 53 is a square of a fraction? But that would not make sense here.

Alternatively, maybe I need to check if there's a mistake in the problem? Like, maybe it was supposed to be sqrt(50) or something else? But the problem says sqrt(53).

Alternatively, maybe I need to check if 53 is a perfect square in some other way? But no, as we saw, it's between 7^2 and 8^2.

Therefore, I think my conclusion is correct. The square root of 53 is already in its simplest radical form.

But just to make sure, let me check an example. For example, sqrt(12) can be simplified to 2*sqrt(3) because 12=4*3. But since 53 has no such factors, it can't be simplified.

Another example: sqrt(18) is 3*sqrt(2). Similarly, sqrt(50) is 5*sqrt(2). But sqrt(53) can't be broken down.

Therefore, I think the answer is sqrt(53).

Wait, but maybe the question is expecting me to write it as something else? Let me check with another method. Let me try prime factorization of 53. Since 53 is prime, its prime factorization is just 53. Therefore, sqrt(53) is sqrt(53), which can't be simplified.

Alternatively, if I write it as sqrt(53) = sqrt(1*53) = sqrt(1)*sqrt(53) = 1*sqrt(53) = sqrt(53). So that's the same thing.

Therefore, there's no way to simplify it further.

Therefore, I think the answer is sqrt(53).

But just to be thorough, let me check an online source or something. Wait, I can't actually access external resources, but I can recall that 53 is indeed a prime number. For example, primes between 1-100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, etc. So yes, 53 is a prime number.

Therefore, sqrt(53) cannot be simplified.

So, the simplest radical form is just sqrt(53).

Therefore, the answer is \boxed{\sqrt{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" means. A square root is in simplest radical form when the number under the square root (the radicand) has no perfect square factors other than 1. This means we should check whether 53 can be factored into a product that includes a perfect square.

---

### Step 1: Prime Factorization of 53

We start by factoring 53. Since 53 is an odd number, it is not divisible by 2. It is not divisible by 3 (since $5 + 3 = 8$, which is not divisible by 3), nor by 5 (it doesn't end in 0 or 5). Testing divisibility by 7, we find that $7 \times 7 = 49$ and $7 \times 8 = 56$, so 53 is not divisible by 7. Continuing this process, we find that 53 is not divisible by any prime number less than or equal to $\sqrt{53} \approx 7.28$. Therefore, **53 is a prime number**.

---

### Step 2: Check for Perfect Square Factors

Since 53 is prime, its only factors are 1 and itself. This means that the only perfect square factor of 53 is 1. Therefore, $\sqrt{53}$ cannot be simplified further by factoring out any perfect square.

---

### Step 3: Confirm That 53 Is Not a Perfect Square

To confirm, we note that $7^2 = 49$ and $8^2 = 64$, so 53 lies between these two perfect squares. Since it is not a perfect square, $\sqrt{53}$ is irrational and cannot be expressed as a simplified radical with a smaller radicand.

---

### Final Conclusion

Because 53 is a prime number and has no perfect square factors other than 1, the square root of 53 is already in its simplest radical form.

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

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