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 start by recalling 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^\circ - x $.

But the problem says the interior angle is 6.5 times the exterior angle. So, according to that, we can write the equation:

$ 180 - x = 6.5x $

Wait, let me check that again. If the interior angle is 6.5 times the exterior angle, then yes, $ \text{Interior} = 6.5 \times \text{Exterior} $. So substituting the expressions, that would be $ 180 - x = 6.5x $.

Let me solve this equation for $ x $. Adding $ x $ to both sides:

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

Then, dividing both sides by 7.5:

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

Hmm, let me compute that. Let's see, 7.5 times 24 is 180 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 $ \frac{360^\circ}{n} $, where $ n $ is the number of sides.

So, since we found that each exterior angle is 24 degrees, we can set up the equation:

$ \frac{360}{n} = 24 $

Solving for $ n $:

Multiply both sides by $ n $:

$ 360 = 24n $

Then divide both sides by 24:

$ n = \frac{360}{24} $

Calculating that, 24 times 15 is 360, right? 24 * 10 = 240, 24 * 5 = 120, so 240 + 120 = 360. Therefore, $ n = 15 $.

Wait, so the polygon has 15 sides? Let me verify that.

Let me check the exterior angle first. If there are 15 sides, each exterior angle is 360/15 = 24 degrees. Then the interior angle would be 180 - 24 = 156 degrees. Now, let's check if 156 is 6.5 times 24.

Calculating 6.5 * 24: 6 * 24 = 144, 0.5 * 24 = 12, so 144 + 12 = 156. Yep, that's correct.

Therefore, the number of sides is 15.

But just to make sure I didn't make any mistakes in my reasoning. Let me recap:

1. Let the exterior angle be x. Then interior angle is 180 - x.
2. Given that interior angle is 6.5x, so 180 - x = 6.5x.
3. Solving gives x = 24 degrees.
4. Since the sum of exterior angles is 360, number of sides is 360 / x = 360 / 24 = 15.

Everything checks out.

Alternatively, I can approach this using the formula for interior angles. The measure of each interior angle of a regular polygon is given by:

$ \frac{(n - 2) \times 180}{n} $

And the measure of each exterior angle is:

$ \frac{360}{n} $

Given that interior angle is 6.5 times the exterior angle:

$ \frac{(n - 2) \times 180}{n} = 6.5 \times \frac{360}{n} $

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

$ (n - 2) \times 180 = 6.5 \times 360 $

Calculate the right-hand side: 6.5 * 360. Let me compute that.

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

Left-hand side: 180(n - 2) = 2340.

Divide both sides by 180:

n - 2 = 2340 / 180

Simplify 2340 / 180: Let's divide numerator and denominator by 10: 234 / 18. Then divide numerator and denominator by 9: 26 / 2 = 13. Wait, 18*13 is 234. So 2340 / 180 = 13. Therefore:

n - 2 = 13 => n = 15.

Same result. So that's consistent.

Therefore, the number of sides is 15.

So, I think that's solid. Both methods lead to the same answer, and verification shows that the interior angle is indeed 6.5 times the exterior angle.

Another way to think about it: The ratio of interior to exterior angle is 6.5:1. Since they are supplementary, their sum is 180. So if I think of it as parts, 6.5 parts interior and 1 part exterior, total parts = 7.5 parts.

So each part is 180 / 7.5 = 24 degrees. Which is the exterior angle. Then same as before.

So same answer.

Therefore, I can be confident that the polygon has 15 sides.

**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 begin by analyzing the relationship between interior and exterior angles in a regular polygon.

---

### Step 1: Define the Angles

In any polygon:
- The sum of an **interior angle** and an **exterior angle** is always $180^\circ$, since they are **supplementary angles**.
- Let the measure of **one exterior angle** be $ x $.
- Then, the **interior angle** is $ 180^\circ - x $.

Given:
$$
\text{Interior angle} = 6.5 \times \text{Exterior angle}
$$
$$
180 - x = 6.5x
$$

---

### Step 2: Solve for the Exterior Angle

Solving the equation:

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

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

---

### Step 3: Use the Relationship Between Exterior Angles and the Number of Sides

In any regular polygon, the sum of the **exterior angles** is always $360^\circ$, and since the polygon is regular, all exterior angles are equal. Thus, the measure of each exterior angle is:

$$
\text{Exterior angle} = \frac{360^\circ}{n}
$$

We already calculated the exterior angle to be $24^\circ$, so:

$$
\frac{360}{n} = 24
$$

Now solve for $n$:

$$
n = \frac{360}{24} = 15
$$

---

### Step 4: Verification

Let’s check that these angles satisfy the original condition:

- Exterior angle = $24^\circ$
- Interior angle = $180 - 24 = 156^\circ$
- Is $156$ equal to $6.5 \times 24$?

$$
6.5 \times 24 = 156
$$

Yes, the relationship is satisfied.

---

### 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 start by recalling 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^\circ - x $.

But the problem says the interior angle is 6.5 times the exterior angle. So, according to that, we can write the equation:

$ 180 - x = 6.5x $

Wait, let me check that again. If the interior angle is 6.5 times the exterior angle, then yes, $ \text{Interior} = 6.5 \times \text{Exterior} $. So substituting the expressions, that would be $ 180 - x = 6.5x $.

Let me solve this equation for $ x $. Adding $ x $ to both sides:

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

Then, dividing both sides by 7.5:

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

Hmm, let me compute that. Let's see, 7.5 times 24 is 180 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 $ \frac{360^\circ}{n} $, where $ n $ is the number of sides.

So, since we found that each exterior angle is 24 degrees, we can set up the equation:

$ \frac{360}{n} = 24 $

Solving for $ n $:

Multiply both sides by $ n $:

$ 360 = 24n $

Then divide both sides by 24:

$ n = \frac{360}{24} $

Calculating that, 24 times 15 is 360, right? 24 * 10 = 240, 24 * 5 = 120, so 240 + 120 = 360. Therefore, $ n = 15 $.

Wait, so the polygon has 15 sides? Let me verify that.

Let me check the exterior angle first. If there are 15 sides, each exterior angle is 360/15 = 24 degrees. Then the interior angle would be 180 - 24 = 156 degrees. Now, let's check if 156 is 6.5 times 24.

Calculating 6.5 * 24: 6 * 24 = 144, 0.5 * 24 = 12, so 144 + 12 = 156. Yep, that's correct.

Therefore, the number of sides is 15.

But just to make sure I didn't make any mistakes in my reasoning. Let me recap:

1. Let the exterior angle be x. Then interior angle is 180 - x.
2. Given that interior angle is 6.5x, so 180 - x = 6.5x.
3. Solving gives x = 24 degrees.
4. Since the sum of exterior angles is 360, number of sides is 360 / x = 360 / 24 = 15.

Everything checks out.

Alternatively, I can approach this using the formula for interior angles. The measure of each interior angle of a regular polygon is given by:

$ \frac{(n - 2) \times 180}{n} $

And the measure of each exterior angle is:

$ \frac{360}{n} $

Given that interior angle is 6.5 times the exterior angle:

$ \frac{(n - 2) \times 180}{n} = 6.5 \times \frac{360}{n} $

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

$ (n - 2) \times 180 = 6.5 \times 360 $

Calculate the right-hand side: 6.5 * 360. Let me compute that.

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

Left-hand side: 180(n - 2) = 2340.

Divide both sides by 180:

n - 2 = 2340 / 180

Simplify 2340 / 180: Let's divide numerator and denominator by 10: 234 / 18. Then divide numerator and denominator by 9: 26 / 2 = 13. Wait, 18*13 is 234. So 2340 / 180 = 13. Therefore:

n - 2 = 13 => n = 15.

Same result. So that's consistent.

Therefore, the number of sides is 15.

So, I think that's solid. Both methods lead to the same answer, and verification shows that the interior angle is indeed 6.5 times the exterior angle.

Another way to think about it: The ratio of interior to exterior angle is 6.5:1. Since they are supplementary, their sum is 180. So if I think of it as parts, 6.5 parts interior and 1 part exterior, total parts = 7.5 parts.

So each part is 180 / 7.5 = 24 degrees. Which is the exterior angle. Then same as before.

So same answer.

Therefore, I can be confident that the polygon has 15 sides.

**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 begin by analyzing the relationship between interior and exterior angles in a regular polygon.

---

### Step 1: Define the Angles

In any polygon:
- The sum of an **interior angle** and an **exterior angle** is always $180^\circ$, since they are **supplementary angles**.
- Let the measure of **one exterior angle** be $ x $.
- Then, the **interior angle** is $ 180^\circ - x $.

Given:
$$
\text{Interior angle} = 6.5 \times \text{Exterior angle}
$$
$$
180 - x = 6.5x
$$

---

### Step 2: Solve for the Exterior Angle

Solving the equation:

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

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

---

### Step 3: Use the Relationship Between Exterior Angles and the Number of Sides

In any regular polygon, the sum of the **exterior angles** is always $360^\circ$, and since the polygon is regular, all exterior angles are equal. Thus, the measure of each exterior angle is:

$$
\text{Exterior angle} = \frac{360^\circ}{n}
$$

We already calculated the exterior angle to be $24^\circ$, so:

$$
\frac{360}{n} = 24
$$

Now solve for $n$:

$$
n = \frac{360}{24} = 15
$$

---

### Step 4: Verification

Let’s check that these angles satisfy the original condition:

- Exterior angle = $24^\circ$
- Interior angle = $180 - 24 = 156^\circ$
- Is $156$ equal to $6.5 \times 24$?

$$
6.5 \times 24 = 156
$$

Yes, the relationship is satisfied.

---

### 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... Right, this is related to solving the congruence equation.

First, let me recall that the inverse of a number modulo n exists only if the number and n are coprime. So I should check if 997 and 1000 are coprime. To do that, I can compute their greatest common divisor (gcd). Since 1000 factors into 2^3 * 5^3, right? And 997... Wait, 997 is a prime number? Let me check. 997... Well, I know that 31^2 is 961 and 32^2 is 1024, so primes up to 31 need to be checked. Let's see: 997 divided by 2? No, it's odd. Divided by 3? 3*332 is 996, so 997-996=1, so remainder 1. Divided by 5? Ends with 7, so no. 7? 7*142 is 994, so 997-994=3, remainder 3. 11? Let me do the alternating sum: 9 - 9 + 7 = 7, not divisible by 11. 13? Let me try 13*76 = 988, 997-988=9, not divisible by 13. 17? 17*58 is 986, 997-986=11, not divisible by 17. 19? 19*52=988, same as before. 23? 23*43=989, 997-989=8, nope. 29? 29*34=986, again 11 left. 31? 31*32=992, 997-992=5. So no, 997 is not divisible by any primes up to its square root, so 997 is prime. Therefore, since 997 is prime and 1000 is 2^3*5^3, and 997 is not divisible by 2 or 5, so gcd(997,1000)=1. Therefore, the inverse exists.

Okay, so how do I find the inverse? Well, one method is to use the Extended Euclidean Algorithm. Let me recall how that works. The Extended Euclidean Algorithm not only finds the gcd of two numbers but also finds integers x and y such that ax + by = gcd(a,b). Since we know that gcd(997, 1000) is 1, this will give me x such that 997x + 1000y = 1. Then, x modulo 1000 will be the inverse of 997 modulo 1000.

So let me set up the algorithm. Let me let a = 1000 and b = 997, since the algorithm is usually presented with a > b. So:

First step: divide 1000 by 997.

1000 = 1*997 + 3.

So, remainder r1 = 3.

Now, divide 997 by r1 = 3.

997 = 332*3 + 1.

Remainder r2 = 1.

Then divide 3 by r2 = 1.

3 = 3*1 + 0.

Since the remainder is 0, the algorithm stops. The last non-zero remainder is 1, which is the gcd.

Now, working backwards to express 1 as a linear combination of 1000 and 997.

Starting from the second last equation: 1 = 997 - 332*3.

But 3 is from the first equation: 3 = 1000 - 1*997.

Substitute that into the above equation:

1 = 997 - 332*(1000 - 1*997)

Let me compute 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:

1 = 333*997 - 332*1000

Therefore, modulo 1000, the term -332*1000 is congruent to 0, so we have:

1 ≡ 333*997 mod 1000

Therefore, 333 is the inverse of 997 modulo 1000? Wait, but let me check.

Wait, 333*997 mod 1000. Let me compute that.

Alternatively, perhaps I made a miscalculation in the coefficients. Let me verify.

So, from the equation:

1 = 333*997 - 332*1000

Therefore, 333*997 ≡ 1 mod 1000. Therefore, the inverse is 333?

But let me check with multiplication. Let's compute 997 * 333. Well, 997 is -3 mod 1000. Wait, 997 ≡ -3 mod 1000, so 997 * 333 ≡ (-3)*333 mod 1000.

Compute (-3)*333 = -999 ≡ 1 mod 1000. Because -999 + 1000 = 1. Yes! So that works. Therefore, 333 is indeed the inverse.

But let me confirm with actual multiplication.

Let me compute 997 * 333.

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

Subtract 3*333 from that? Wait, because 997 is 1000 - 3.

So 997 * 333 = (1000 - 3) * 333 = 1000*333 - 3*333.

As above, 1000*333 is 333,000. 3*333 is 999. Therefore, 333,000 - 999 = 332,001.

Now, 332,001 modulo 1000 is the last three digits: 001. So 1. Therefore, indeed, 997*333 ≡ 1 mod 1000.

So the inverse is 333.

Wait, but the question says "Express your answer as an integer from 0 to 999." So 333 is between 0 and 999, so that's the answer?

But maybe I should check if there's another way to compute this inverse? Maybe using another method? Let me try another approach, just to verify.

Another way is to note that since 997 ≡ -3 mod 1000, so we need to find the inverse of -3 mod 1000. Which is equivalent to finding the inverse of 3 mod 1000 and then multiplying by -1.

So, if I can find the inverse of 3 mod 1000, then multiply by -1, that would be the inverse of -3 mod 1000.

So let's find the inverse of 3 mod 1000.

So, find x such that 3x ≡ 1 mod 1000.

We can solve this equation.

One way is to use the Extended Euclidean Algorithm again.

Let me do that.

Find gcd(3, 1000).

1000 divided by 3 is 333 with remainder 1.

So 1000 = 3*333 + 1.

Then, 3 = 1*3 + 0.

Therefore, gcd is 1.

Working backwards:

1 = 1000 - 3*333

Therefore, 1 = (-333)*3 + 1*1000

So, the inverse of 3 mod 1000 is -333 mod 1000.

-333 mod 1000 is 1000 - 333 = 667.

Therefore, inverse of 3 mod 1000 is 667.

Therefore, inverse of -3 mod 1000 is -667 mod 1000, which is 1000 - 667 = 333.

Therefore, same result. So 333 is the inverse. Therefore, that confirms the previous result.

Alternatively, maybe I can use another approach, like solving the congruence 997x ≡ 1 mod 1000. Which is equivalent to -3x ≡ 1 mod 1000, so 3x ≡ -1 mod 1000, which is 3x ≡ 999 mod 1000.

So, solving 3x ≡ 999 mod 1000. Divide both sides by gcd(3,1000)=1, so x ≡ 999 * 3^{-1} mod 1000. Which again, since 3^{-1} is 667, so x ≡ 999 * 667 mod 1000.

But 999 ≡ -1 mod 1000, so x ≡ (-1)*667 ≡ -667 ≡ 333 mod 1000. Same answer.

Alternatively, maybe I can think of solving 3x ≡ 999 mod 1000.

Divide both sides by... Well, 3 divides into 999 exactly 333 times (since 3*333=999). So 3x ≡ 3*333 mod 1000. Then, divide both sides by 3: x ≡ 333 mod 1000/ gcd(3,1000)= 333 mod 1000. So same result.

So, again, same answer.

Alternatively, maybe trial and error? Although that might be tedious. But given that 333 works, but let me check if there's another way.

Alternatively, maybe using Euler's theorem? Since 997 and 1000 are coprime, Euler's theorem says that 997^φ(1000) ≡ 1 mod 1000. φ(1000) is φ(2^3 * 5^3) = 1000*(1 - 1/2)*(1 - 1/5) = 1000*(1/2)*(4/5) = 1000*(2/5) = 400. So φ(1000) is 400. Therefore, 997^400 ≡ 1 mod 1000. Therefore, 997^{-1} ≡ 997^{399} mod 1000. But computing 997^{399} mod 1000 seems difficult. Maybe not helpful here.

Alternatively, maybe using successive squaring? But again, perhaps not the easiest way here.

Alternatively, maybe notice that 997 ≡ -3 mod 1000, so inverse is -3^{-1} mod 1000. As above.

Alternatively, perhaps note that 3 * 667 = 2001. Which is 2*1000 + 1. Therefore, 3*667 ≡ 1 mod 1000. Therefore, inverse of 3 is 667. Then inverse of -3 is -667 ≡ 333.

Alternatively, maybe I can check 333 * 997.

As before, 333 * 997 = 333*(1000 - 3) = 333,000 - 999 = 332,001. Then, mod 1000, that's 001. So yes, 1.

Therefore, 333 is correct.

But maybe the answer is different? Wait, let me check with another method.

Suppose I want to solve 997x ≡ 1 mod 1000. Let me write this as 997x - 1000y = 1. So solving for integers x and y.

But we already did that with the Extended Euclidean Algorithm and found x=333.

Alternatively, maybe I can solve it step by step. Let me try to write equations.

Let me think of this as 997x ≡ 1 mod 1000.

Which is same as 997x - 1 = 1000y for some integer y.

But maybe trying to find x such that 997x ≡ 1 mod 1000.

Alternatively, since mod 1000, we can write this as solving 997x ≡ 1 mod 8 and mod 125, then use Chinese Remainder Theorem? Because 1000 = 8*125, and 8 and 125 are coprime.

So maybe solving the congruence modulo 8 and modulo 125, then combine the solutions.

Let me try that approach.

First, solve 997x ≡ 1 mod 8.

Compute 997 mod 8. 8*124 = 992, so 997 - 992 = 5. Therefore, 997 ≡ 5 mod 8.

Thus, equation becomes 5x ≡ 1 mod 8.

Find x such that 5x ≡ 1 mod 8.

Multiplicative inverse of 5 mod 8. Let me compute:

5*1 = 5 mod 8

5*2 = 10 ≡ 2 mod 8

5*3 = 15 ≡ 7 mod 8

5*5 = 25 ≡ 1 mod 8.

Wait, 5*5 = 25, 25-3*8=1, so yes. Therefore, inverse is 5 mod 8. Therefore, x ≡ 5 mod 8.

So x ≡ 5 mod 8.

Now, solve the congruence 997x ≡ 1 mod 125.

Compute 997 mod 125.

First, 125*7 = 875. 997 - 875 = 122. 122 is less than 125. Therefore, 997 ≡ 122 mod 125.

So equation becomes 122x ≡ 1 mod 125.

Need to find inverse of 122 mod 125.

Alternatively, since 122 ≡ -3 mod 125. So, inverse of -3 mod 125 is same as inverse of 122 mod 125.

So we can write -3x ≡ 1 mod 125, which is equivalent to 3x ≡ -1 mod 125, or 3x ≡ 124 mod 125.

So, solving 3x ≡ 124 mod 125.

Find x.

We can use the Extended Euclidean Algorithm again.

Find gcd(3, 125).

125 = 3*41 + 2

3 = 2*1 + 1

2 = 1*2 + 0

So gcd is 1.

Now, backtracking:

1 = 3 - 2*1

But 2 = 125 - 3*41

Therefore, 1 = 3 - (125 - 3*41)*1 = 3 - 125 + 3*41 = 3*(1 + 41) - 125 = 3*42 - 125*1

Therefore, 1 = 3*42 - 125*1

Therefore, 3*42 ≡ 1 mod 125. Therefore, inverse of 3 mod 125 is 42.

Therefore, going back to equation 3x ≡ 124 mod 125, multiply both sides by 42:

x ≡ 124 * 42 mod 125

Compute 124 * 42:

Note that 124 ≡ -1 mod 125, so -1 * 42 = -42 mod 125.

Which is equivalent to 125 - 42 = 83 mod 125.

Therefore, x ≡ 83 mod 125.

Therefore, the solutions for the original congruence modulo 125 is x ≡ 83 mod 125.

So now, we have two congruences:

x ≡ 5 mod 8

x ≡ 83 mod 125

We need to find x such that both are satisfied, and then x mod 1000 will be our answer.

So let me solve these two congruences.

Let me write x = 125k + 83 for some integer k. Then, substitute into first congruence:

125k + 83 ≡ 5 mod 8

Compute 125 mod 8: 125 / 8 = 15*8 = 120, remainder 5. So 125 ≡ 5 mod 8.

Similarly, 83 mod 8: 8*10=80, 83 - 80=3. So 83 ≡ 3 mod 8.

So equation becomes:

5k + 3 ≡ 5 mod 8

Subtract 3:

5k ≡ 2 mod 8

So need to solve 5k ≡ 2 mod 8.

Again, inverse of 5 mod 8 is 5, as before, since 5*5=25≡1 mod8.

Multiply both sides by 5:

k ≡ 2*5 mod 8

k ≡ 10 mod 8

Which is k ≡ 2 mod 8.

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

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

Therefore, solutions are x ≡ 333 mod 1000.

Therefore, x = 333 is the solution modulo 1000.

Therefore, the inverse is 333.

So, same answer as before. Therefore, regardless of the method, the answer seems to be 333.

Let me check once more with another approach.

Let me try to compute 997^{-1} mod 1000 by using the fact that 997 ≡ -3 mod 1000. Therefore, we need to find -3^{-1} mod 1000.

So, if I can find the inverse of 3 mod 1000, which we found was 667, so inverse of -3 is -667 mod 1000, which is 1000 - 667 = 333.

Yes, same answer.

Alternatively, perhaps using the fact that 997 ≡ -3 mod 1000, so 997^2 ≡ (-3)^2 = 9 mod 1000. 997^3 ≡ -3^3 = -27 mod 1000. 997^4 ≡ 81 mod 1000. 997^5 ≡ -243 mod 1000. 997^6 ≡ 729 mod 1000. 997^7 ≡ -2187 mod 1000. Which is -2187 + 3*1000 = 813 mod 1000. 997^8 ≡ -3*813 = -2439 mod 1000. Which is -2439 + 3*1000 = 561.

Hmm, maybe this is getting too complicated, but perhaps we can notice that 997^something... But maybe it's not helpful here.

Alternatively, maybe use the fact that 333 is the inverse, and since the problem checks out with multiple methods, perhaps it is correct.

But just to be thorough, perhaps I can compute 997 * 333 mod 1000 in another way.

Note that 997 * 333 = (1000 - 3) * 333 = 1000*333 - 3*333.

Modulo 1000, the 1000*333 term disappears, leaving -3*333 mod 1000.

Calculating 3*333 = 999. So -999 mod 1000 is 1.

Therefore, indeed, 997 * 333 ≡ 1 mod 1000.

So, multiple confirmations.

Therefore, I think 333 is the correct answer.

But just to make sure I didn't make any arithmetic errors in my steps.

Let me recap:

Since 997 ≡ -3 mod 1000, solving for inverse is solving -3x ≡ 1 mod 1000 => 3x ≡ -1 mod 1000 => 3x ≡ 999 mod 1000.

Then, solving 3x ≡ 999 mod 1000.

Divided both sides by 3: x ≡ 333 mod 1000, since 999 / 3 = 333.

Wait, that's even more straightforward. Since 3*333 = 999, so 3x ≡ 999 mod 1000 => x ≡ 333 mod 1000.

That's another way to see it.

Therefore, that's a direct solution.

Therefore, since 3 divides 999, and gcd(3,1000)=1, so we can divide both sides by 3.

Therefore, x ≡ 333 mod 1000.

Therefore, that's the answer.

Therefore, I think that 333 is correct.

Hence, the answer is 333.

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

To compute the inverse of $ 997 $ modulo $ 1000 $, we aim to find an integer $ x $ such that:

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

---

### Step 1: Check if an inverse exists

A number has an inverse modulo $ n $ if and only if it is coprime to $ n $. That is, we must have $ \gcd(997, 1000) = 1 $.

The prime factorization of $ 1000 $ is:

$$
1000 = 2^3 \cdot 5^3
$$

Since $ 997 $ is a prime number (and not divisible by 2 or 5), it is coprime to 1000. Hence, an inverse exists.

---

### Step 2: Simplify the congruence

Note that:

$$
997 \equiv -3 \pmod{1000}
$$

So the congruence becomes:

$$
(-3)x \equiv 1 \pmod{1000}
\Rightarrow 3x \equiv -1 \pmod{1000}
\Rightarrow 3x \equiv 999 \pmod{1000}
$$

---

### Step 3: Solve the congruence

We now solve the linear congruence:

$$
3x \equiv 999 \pmod{1000}
$$

Dividing both sides by 3 (since $ \gcd(3, 1000) = 1 $), we get:

$$
x \equiv \frac{999}{3} \pmod{1000} \Rightarrow x \equiv 333 \pmod{1000}
$$

Therefore, the solution is:

$$
x = 333
$$

---

### Step 4: Verification

To verify, compute:

$$
997 \times 333 = (1000 - 3) \times 333 = 1000 \cdot 333 - 3 \cdot 333
$$

$$
= 333000 - 999 = 332001
$$

Then:

$$
332001 \mod 1000 = 1
$$

This confirms that:

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

---

### 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... Right, this is related to solving the congruence equation.

First, let me recall that the inverse of a number modulo n exists only if the number and n are coprime. So I should check if 997 and 1000 are coprime. To do that, I can compute their greatest common divisor (gcd). Since 1000 factors into 2^3 * 5^3, right? And 997... Wait, 997 is a prime number? Let me check. 997... Well, I know that 31^2 is 961 and 32^2 is 1024, so primes up to 31 need to be checked. Let's see: 997 divided by 2? No, it's odd. Divided by 3? 3*332 is 996, so 997-996=1, so remainder 1. Divided by 5? Ends with 7, so no. 7? 7*142 is 994, so 997-994=3, remainder 3. 11? Let me do the alternating sum: 9 - 9 + 7 = 7, not divisible by 11. 13? Let me try 13*76 = 988, 997-988=9, not divisible by 13. 17? 17*58 is 986, 997-986=11, not divisible by 17. 19? 19*52=988, same as before. 23? 23*43=989, 997-989=8, nope. 29? 29*34=986, again 11 left. 31? 31*32=992, 997-992=5. So no, 997 is not divisible by any primes up to its square root, so 997 is prime. Therefore, since 997 is prime and 1000 is 2^3*5^3, and 997 is not divisible by 2 or 5, so gcd(997,1000)=1. Therefore, the inverse exists.

Okay, so how do I find the inverse? Well, one method is to use the Extended Euclidean Algorithm. Let me recall how that works. The Extended Euclidean Algorithm not only finds the gcd of two numbers but also finds integers x and y such that ax + by = gcd(a,b). Since we know that gcd(997, 1000) is 1, this will give me x such that 997x + 1000y = 1. Then, x modulo 1000 will be the inverse of 997 modulo 1000.

So let me set up the algorithm. Let me let a = 1000 and b = 997, since the algorithm is usually presented with a > b. So:

First step: divide 1000 by 997.

1000 = 1*997 + 3.

So, remainder r1 = 3.

Now, divide 997 by r1 = 3.

997 = 332*3 + 1.

Remainder r2 = 1.

Then divide 3 by r2 = 1.

3 = 3*1 + 0.

Since the remainder is 0, the algorithm stops. The last non-zero remainder is 1, which is the gcd.

Now, working backwards to express 1 as a linear combination of 1000 and 997.

Starting from the second last equation: 1 = 997 - 332*3.

But 3 is from the first equation: 3 = 1000 - 1*997.

Substitute that into the above equation:

1 = 997 - 332*(1000 - 1*997)

Let me compute 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:

1 = 333*997 - 332*1000

Therefore, modulo 1000, the term -332*1000 is congruent to 0, so we have:

1 ≡ 333*997 mod 1000

Therefore, 333 is the inverse of 997 modulo 1000? Wait, but let me check.

Wait, 333*997 mod 1000. Let me compute that.

Alternatively, perhaps I made a miscalculation in the coefficients. Let me verify.

So, from the equation:

1 = 333*997 - 332*1000

Therefore, 333*997 ≡ 1 mod 1000. Therefore, the inverse is 333?

But let me check with multiplication. Let's compute 997 * 333. Well, 997 is -3 mod 1000. Wait, 997 ≡ -3 mod 1000, so 997 * 333 ≡ (-3)*333 mod 1000.

Compute (-3)*333 = -999 ≡ 1 mod 1000. Because -999 + 1000 = 1. Yes! So that works. Therefore, 333 is indeed the inverse.

But let me confirm with actual multiplication.

Let me compute 997 * 333.

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

Subtract 3*333 from that? Wait, because 997 is 1000 - 3.

So 997 * 333 = (1000 - 3) * 333 = 1000*333 - 3*333.

As above, 1000*333 is 333,000. 3*333 is 999. Therefore, 333,000 - 999 = 332,001.

Now, 332,001 modulo 1000 is the last three digits: 001. So 1. Therefore, indeed, 997*333 ≡ 1 mod 1000.

So the inverse is 333.

Wait, but the question says "Express your answer as an integer from 0 to 999." So 333 is between 0 and 999, so that's the answer?

But maybe I should check if there's another way to compute this inverse? Maybe using another method? Let me try another approach, just to verify.

Another way is to note that since 997 ≡ -3 mod 1000, so we need to find the inverse of -3 mod 1000. Which is equivalent to finding the inverse of 3 mod 1000 and then multiplying by -1.

So, if I can find the inverse of 3 mod 1000, then multiply by -1, that would be the inverse of -3 mod 1000.

So let's find the inverse of 3 mod 1000.

So, find x such that 3x ≡ 1 mod 1000.

We can solve this equation.

One way is to use the Extended Euclidean Algorithm again.

Let me do that.

Find gcd(3, 1000).

1000 divided by 3 is 333 with remainder 1.

So 1000 = 3*333 + 1.

Then, 3 = 1*3 + 0.

Therefore, gcd is 1.

Working backwards:

1 = 1000 - 3*333

Therefore, 1 = (-333)*3 + 1*1000

So, the inverse of 3 mod 1000 is -333 mod 1000.

-333 mod 1000 is 1000 - 333 = 667.

Therefore, inverse of 3 mod 1000 is 667.

Therefore, inverse of -3 mod 1000 is -667 mod 1000, which is 1000 - 667 = 333.

Therefore, same result. So 333 is the inverse. Therefore, that confirms the previous result.

Alternatively, maybe I can use another approach, like solving the congruence 997x ≡ 1 mod 1000. Which is equivalent to -3x ≡ 1 mod 1000, so 3x ≡ -1 mod 1000, which is 3x ≡ 999 mod 1000.

So, solving 3x ≡ 999 mod 1000. Divide both sides by gcd(3,1000)=1, so x ≡ 999 * 3^{-1} mod 1000. Which again, since 3^{-1} is 667, so x ≡ 999 * 667 mod 1000.

But 999 ≡ -1 mod 1000, so x ≡ (-1)*667 ≡ -667 ≡ 333 mod 1000. Same answer.

Alternatively, maybe I can think of solving 3x ≡ 999 mod 1000.

Divide both sides by... Well, 3 divides into 999 exactly 333 times (since 3*333=999). So 3x ≡ 3*333 mod 1000. Then, divide both sides by 3: x ≡ 333 mod 1000/ gcd(3,1000)= 333 mod 1000. So same result.

So, again, same answer.

Alternatively, maybe trial and error? Although that might be tedious. But given that 333 works, but let me check if there's another way.

Alternatively, maybe using Euler's theorem? Since 997 and 1000 are coprime, Euler's theorem says that 997^φ(1000) ≡ 1 mod 1000. φ(1000) is φ(2^3 * 5^3) = 1000*(1 - 1/2)*(1 - 1/5) = 1000*(1/2)*(4/5) = 1000*(2/5) = 400. So φ(1000) is 400. Therefore, 997^400 ≡ 1 mod 1000. Therefore, 997^{-1} ≡ 997^{399} mod 1000. But computing 997^{399} mod 1000 seems difficult. Maybe not helpful here.

Alternatively, maybe using successive squaring? But again, perhaps not the easiest way here.

Alternatively, maybe notice that 997 ≡ -3 mod 1000, so inverse is -3^{-1} mod 1000. As above.

Alternatively, perhaps note that 3 * 667 = 2001. Which is 2*1000 + 1. Therefore, 3*667 ≡ 1 mod 1000. Therefore, inverse of 3 is 667. Then inverse of -3 is -667 ≡ 333.

Alternatively, maybe I can check 333 * 997.

As before, 333 * 997 = 333*(1000 - 3) = 333,000 - 999 = 332,001. Then, mod 1000, that's 001. So yes, 1.

Therefore, 333 is correct.

But maybe the answer is different? Wait, let me check with another method.

Suppose I want to solve 997x ≡ 1 mod 1000. Let me write this as 997x - 1000y = 1. So solving for integers x and y.

But we already did that with the Extended Euclidean Algorithm and found x=333.

Alternatively, maybe I can solve it step by step. Let me try to write equations.

Let me think of this as 997x ≡ 1 mod 1000.

Which is same as 997x - 1 = 1000y for some integer y.

But maybe trying to find x such that 997x ≡ 1 mod 1000.

Alternatively, since mod 1000, we can write this as solving 997x ≡ 1 mod 8 and mod 125, then use Chinese Remainder Theorem? Because 1000 = 8*125, and 8 and 125 are coprime.

So maybe solving the congruence modulo 8 and modulo 125, then combine the solutions.

Let me try that approach.

First, solve 997x ≡ 1 mod 8.

Compute 997 mod 8. 8*124 = 992, so 997 - 992 = 5. Therefore, 997 ≡ 5 mod 8.

Thus, equation becomes 5x ≡ 1 mod 8.

Find x such that 5x ≡ 1 mod 8.

Multiplicative inverse of 5 mod 8. Let me compute:

5*1 = 5 mod 8

5*2 = 10 ≡ 2 mod 8

5*3 = 15 ≡ 7 mod 8

5*5 = 25 ≡ 1 mod 8.

Wait, 5*5 = 25, 25-3*8=1, so yes. Therefore, inverse is 5 mod 8. Therefore, x ≡ 5 mod 8.

So x ≡ 5 mod 8.

Now, solve the congruence 997x ≡ 1 mod 125.

Compute 997 mod 125.

First, 125*7 = 875. 997 - 875 = 122. 122 is less than 125. Therefore, 997 ≡ 122 mod 125.

So equation becomes 122x ≡ 1 mod 125.

Need to find inverse of 122 mod 125.

Alternatively, since 122 ≡ -3 mod 125. So, inverse of -3 mod 125 is same as inverse of 122 mod 125.

So we can write -3x ≡ 1 mod 125, which is equivalent to 3x ≡ -1 mod 125, or 3x ≡ 124 mod 125.

So, solving 3x ≡ 124 mod 125.

Find x.

We can use the Extended Euclidean Algorithm again.

Find gcd(3, 125).

125 = 3*41 + 2

3 = 2*1 + 1

2 = 1*2 + 0

So gcd is 1.

Now, backtracking:

1 = 3 - 2*1

But 2 = 125 - 3*41

Therefore, 1 = 3 - (125 - 3*41)*1 = 3 - 125 + 3*41 = 3*(1 + 41) - 125 = 3*42 - 125*1

Therefore, 1 = 3*42 - 125*1

Therefore, 3*42 ≡ 1 mod 125. Therefore, inverse of 3 mod 125 is 42.

Therefore, going back to equation 3x ≡ 124 mod 125, multiply both sides by 42:

x ≡ 124 * 42 mod 125

Compute 124 * 42:

Note that 124 ≡ -1 mod 125, so -1 * 42 = -42 mod 125.

Which is equivalent to 125 - 42 = 83 mod 125.

Therefore, x ≡ 83 mod 125.

Therefore, the solutions for the original congruence modulo 125 is x ≡ 83 mod 125.

So now, we have two congruences:

x ≡ 5 mod 8

x ≡ 83 mod 125

We need to find x such that both are satisfied, and then x mod 1000 will be our answer.

So let me solve these two congruences.

Let me write x = 125k + 83 for some integer k. Then, substitute into first congruence:

125k + 83 ≡ 5 mod 8

Compute 125 mod 8: 125 / 8 = 15*8 = 120, remainder 5. So 125 ≡ 5 mod 8.

Similarly, 83 mod 8: 8*10=80, 83 - 80=3. So 83 ≡ 3 mod 8.

So equation becomes:

5k + 3 ≡ 5 mod 8

Subtract 3:

5k ≡ 2 mod 8

So need to solve 5k ≡ 2 mod 8.

Again, inverse of 5 mod 8 is 5, as before, since 5*5=25≡1 mod8.

Multiply both sides by 5:

k ≡ 2*5 mod 8

k ≡ 10 mod 8

Which is k ≡ 2 mod 8.

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

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

Therefore, solutions are x ≡ 333 mod 1000.

Therefore, x = 333 is the solution modulo 1000.

Therefore, the inverse is 333.

So, same answer as before. Therefore, regardless of the method, the answer seems to be 333.

Let me check once more with another approach.

Let me try to compute 997^{-1} mod 1000 by using the fact that 997 ≡ -3 mod 1000. Therefore, we need to find -3^{-1} mod 1000.

So, if I can find the inverse of 3 mod 1000, which we found was 667, so inverse of -3 is -667 mod 1000, which is 1000 - 667 = 333.

Yes, same answer.

Alternatively, perhaps using the fact that 997 ≡ -3 mod 1000, so 997^2 ≡ (-3)^2 = 9 mod 1000. 997^3 ≡ -3^3 = -27 mod 1000. 997^4 ≡ 81 mod 1000. 997^5 ≡ -243 mod 1000. 997^6 ≡ 729 mod 1000. 997^7 ≡ -2187 mod 1000. Which is -2187 + 3*1000 = 813 mod 1000. 997^8 ≡ -3*813 = -2439 mod 1000. Which is -2439 + 3*1000 = 561.

Hmm, maybe this is getting too complicated, but perhaps we can notice that 997^something... But maybe it's not helpful here.

Alternatively, maybe use the fact that 333 is the inverse, and since the problem checks out with multiple methods, perhaps it is correct.

But just to be thorough, perhaps I can compute 997 * 333 mod 1000 in another way.

Note that 997 * 333 = (1000 - 3) * 333 = 1000*333 - 3*333.

Modulo 1000, the 1000*333 term disappears, leaving -3*333 mod 1000.

Calculating 3*333 = 999. So -999 mod 1000 is 1.

Therefore, indeed, 997 * 333 ≡ 1 mod 1000.

So, multiple confirmations.

Therefore, I think 333 is the correct answer.

But just to make sure I didn't make any arithmetic errors in my steps.

Let me recap:

Since 997 ≡ -3 mod 1000, solving for inverse is solving -3x ≡ 1 mod 1000 => 3x ≡ -1 mod 1000 => 3x ≡ 999 mod 1000.

Then, solving 3x ≡ 999 mod 1000.

Divided both sides by 3: x ≡ 333 mod 1000, since 999 / 3 = 333.

Wait, that's even more straightforward. Since 3*333 = 999, so 3x ≡ 999 mod 1000 => x ≡ 333 mod 1000.

That's another way to see it.

Therefore, that's a direct solution.

Therefore, since 3 divides 999, and gcd(3,1000)=1, so we can divide both sides by 3.

Therefore, x ≡ 333 mod 1000.

Therefore, that's the answer.

Therefore, I think that 333 is correct.

Hence, the answer is 333.

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

To compute the inverse of $ 997 $ modulo $ 1000 $, we aim to find an integer $ x $ such that:

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

---

### Step 1: Check if an inverse exists

A number has an inverse modulo $ n $ if and only if it is coprime to $ n $. That is, we must have $ \gcd(997, 1000) = 1 $.

The prime factorization of $ 1000 $ is:

$$
1000 = 2^3 \cdot 5^3
$$

Since $ 997 $ is a prime number (and not divisible by 2 or 5), it is coprime to 1000. Hence, an inverse exists.

---

### Step 2: Simplify the congruence

Note that:

$$
997 \equiv -3 \pmod{1000}
$$

So the congruence becomes:

$$
(-3)x \equiv 1 \pmod{1000}
\Rightarrow 3x \equiv -1 \pmod{1000}
\Rightarrow 3x \equiv 999 \pmod{1000}
$$

---

### Step 3: Solve the congruence

We now solve the linear congruence:

$$
3x \equiv 999 \pmod{1000}
$$

Dividing both sides by 3 (since $ \gcd(3, 1000) = 1 $), we get:

$$
x \equiv \frac{999}{3} \pmod{1000} \Rightarrow x \equiv 333 \pmod{1000}
$$

Therefore, the solution is:

$$
x = 333
$$

---

### Step 4: Verification

To verify, compute:

$$
997 \times 333 = (1000 - 3) \times 333 = 1000 \cdot 333 - 3 \cdot 333
$$

$$
= 333000 - 999 = 332001
$$

Then:

$$
332001 \mod 1000 = 1
$$

This confirms that:

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

---

### 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, let's see. I need to convert the repeating decimal 0.133113311331... into a fraction. The question says it's a repeating decimal, right? So the bar is over 1331, which means the digits 1331 repeat indefinitely. So the decimal is 0.133113311331...

First, I remember that there's a method to convert repeating decimals to fractions. Let me recall how that works. I think you set the decimal equal to a variable, multiply both sides by a power of 10 to move the decimal point so that the repeating parts align, then subtract the equations to eliminate the repeating part. Let me try to apply that here.

Let me denote the repeating decimal as x. So let me write:

Let x = 0.133113311331...

Now, the repeating part is "1331", which is four digits. So the decimal repeats every four digits. Therefore, if I multiply x by 10^4 (which is 10000), I should shift the decimal point four places to the right. Let's try that:

10000x = 1331.133113311331...

But the original x is 0.133113311331... So if I subtract x from 10000x, maybe the repeating parts will cancel out.

So let me write:

10000x = 1331.133113311331...

minus

x = 0.133113311331...

Subtracting the two equations:

10000x - x = 1331.133113311331... - 0.133113311331...

Which gives:

9999x = 1331

So then, solving for x:

x = 1331 / 9999

Wait, but I need to check if this fraction can be simplified. 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.

Let me compute GCD(1331, 9999). First, note that 1331 is a number I recognize. 11^3 is 1331, because 11*11 is 121, 121*11 is 1331. Yes, that's right. So 1331 is 11³.

Now, let's factorize 9999. Let me see. 9999 is 10000 - 1, which factors into (100 - 1)(100 + 1) = 99*101. Wait, 99 is 9*11, and 101 is a prime number. Therefore:

9999 = 9 * 11 * 101.

So 9999 factors into 9 * 11 * 101.

So, 1331 is 11³, and 9999 is 9 * 11 * 101. Therefore, the GCD of 1331 and 9999 is the product of the common prime factors with the lowest exponents. The common prime factor is 11. The exponent of 11 in 1331 is 3, and in 9999 it's 1. So the GCD is 11^1 = 11.

Therefore, we can divide numerator and denominator by 11:

Numerator: 1331 / 11 = let me calculate that. 11*121 is 1331, right? Because 11*120=1320, plus 11 is 1331. So 1331 / 11 = 121.

Denominator: 9999 / 11 = let me compute that. 9999 divided by 11. Let's see, 11*900 = 9900. Then 9999 - 9900 = 99. 99 divided by 11 is 9. So total is 900 + 9 = 909. Therefore, 9999 / 11 = 909.

Therefore, the simplified fraction is 121 / 909.

Wait, but now I need to check if 121 and 909 have any common factors. Let's check GCD(121, 909).

121 is 11². Let's factorize 909. Let's divide 909 by 3: 9 + 0 + 9 = 18, which is divisible by 3. So 909 ÷ 3 = 303. Then 303 ÷ 3 = 101. Therefore, 909 is 3² * 101.

So, prime factors of 121 are 11², and prime factors of 909 are 3² * 101. There are no common factors between 121 and 909, since 11 is not a factor of 909 (since 909 ÷ 11 is 82.63... which isn't integer). Therefore, the fraction 121/909 is in simplest terms.

Therefore, the original decimal 0.13311331... is equal to 121/909, so a is 121 and b is 909. Therefore, a + b is 121 + 909 = 1030.

Wait, let me check again. Let me verify if the initial steps were correct.

Starting with x = 0.13311331..., multiplied by 10^4 = 10000 gives 1331.13311331...

Subtracting x gives 9999x = 1331. So x = 1331 / 9999. Then simplifying. Yes, that seems correct.

Then GCD(1331, 9999) is 11. Dividing numerator and denominator by 11 gives 121 / 909, which reduces to lowest terms. Then 121 + 909 is 1030.

But wait, let me check if 121 and 909 have any common factors. As I said before, 121 is 11². 909 divided by 11 is 82.636..., which is not integer. Let me check 11*82 = 902, 909 - 902 = 7. So remainder 7. Therefore, 11 is not a factor of 909. Therefore, there are no more common factors. Therefore, 121/909 is reduced.

But wait, let me check if 909 has factors of 11? 909 divided by 11 is 82.63... as above, so no. So the fraction is reduced.

Therefore, a + b is 121 + 909 = 1030. Wait, but let me check with another method to be sure.

Alternatively, maybe I made a mistake in the initial setup?

Let me check by converting 121/909 to decimal. Let me perform the division 121 ÷ 909.

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

Multiply numerator and denominator by something? Maybe not. Alternatively, since 1/909 is approximately... Well, perhaps it's easier to check if 121/909 equals 0.1331...

Alternatively, let me do the division 121 divided by 909.

First, note that 909 goes into 121 zero times. Add decimal point, 121.0000...

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 2727 from 3010: 3010 - 2727 = 283. Bring down a zero: 2830.

909 goes into 2830 three times again (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 is where we started after the first step. So the decimal repeats from here. So the decimal expansion is 0.1331331... So 0.1331331..., which is 0.1331 repeating? Wait, but the original decimal was 0.133113311331...

Wait, hold on. Did I get the decimal expansion right?

Wait, according to the division, 121/909 is 0.1331331331..., so the repeating part is "1331", but let me check:

Starting the division:

After decimal:

First step: 121.0000...

First digit: 1, remainder 301.

Second digit: 3, remainder 283.

Third digit: 3, remainder 103.

Fourth digit: 1, remainder 121.

Then it repeats. So the decimal is 0.1331 1331 1331..., so it's 0.133113311331..., which matches the original decimal given in the problem. Therefore, 121/909 is indeed equal to 0.13311331..., so the fraction is correct.

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

Wait, but let me check once more if the initial equation was correct. I set x = 0.13311331..., multiplied by 10000 to get 1331.1331..., then subtracted x to get 9999x = 1331. So x = 1331 / 9999. Then simplified.

But let me check that the decimal is indeed repeating every four digits. The original decimal is 0.133113311331... So the repeating part is "1331", which is four digits. So the method is correct because multiplying by 10^4 shifts the decimal four places, so that subtracting x cancels out the repeating part.

Therefore, all steps seem correct.

But let me check if 1331/9999 reduces to 121/909. 1331 divided by 11 is 121, and 9999 divided by 11 is 909. Then, as we saw, 121 and 909 have no common factors, so yes.

Therefore, the answer is 121 + 909 = 1030.

But wait, just to make sure, let me check if there is another way to think about this. Maybe I made a mistake in the GCD?

Let me recalculate GCD(1331, 9999). Using the Euclidean algorithm:

GCD(9999, 1331). So divide 9999 by 1331.

1331 * 7 = 9317. Subtract that from 9999: 9999 - 9317 = 682.

So GCD(1331, 682).

Now divide 1331 by 682. 682 * 1 = 682. Subtract from 1331: 1331 - 682 = 649.

So GCD(682, 649).

Divide 682 by 649: 649 * 1 = 649. Subtract from 682: 682 - 649 = 33.

GCD(649, 33).

Divide 649 by 33: 33*19 = 627. Subtract: 649 - 627 = 22.

GCD(33, 22).

Divide 33 by 22: 22*1 = 22. Subtract: 33 - 22 = 11.

GCD(22, 11).

Divide 22 by 11: 11*2 = 22. Remainder 0. So GCD is 11.

Therefore, GCD(1331, 9999) is indeed 11. Therefore, the simplification is correct.

So, yes, the answer is 121 + 909 = 1030. Therefore, I think that's correct.

But just to be absolutely thorough, let me check if 121 and 909 have any common factors. As we saw earlier, 121 is 11². Let's check if 11 divides 909. 9 + 0 + 9 = 18, which is not divisible by 11. Wait, the divisibility rule for 11 is alternating sum of digits. Let me apply that.

For 909: digits are 9, 0, 9. Alternating sum: 9 - 0 + 9 = 18. 18 is not divisible by 11, so 909 is not divisible by 11. Therefore, GCD(121, 909) = 1. So yes, 121/909 is reduced.

Therefore, the final answer is 121 + 909 = 1030. Wait, but let me check with another approach. Maybe the decimal is not four digits? Wait, the original decimal is 0.133113311331... So starting from the decimal, the first four digits after the decimal are 1331, then repeating. So the repeating block is four digits. So the method of multiplying by 10^4 is correct. Therefore, the setup was correct.

Alternatively, suppose someone thought that the repeating part is "331" or something else? But the problem says 0.\overline{1331}, which indicates that the bar is over 1331, so the repeating part is four digits: 1, 3, 3, 1. Therefore, the setup is correct.

Therefore, I think the answer is 1030. Therefore, I think that is the correct answer.

**Final Answer**
The value of $a + b$ is \boxed{1030}.

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

---

### Step 1: Let the decimal be represented by a variable

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

---

### Step 2: Multiply by a power of 10 to align the repeating block

Since the repeating block "1331" has 4 digits, we multiply both sides by $10^4 = 10000$ to move the decimal point four places to the right:
$$
10000x = 1331.\overline{1331}
$$

Now subtract the original $x$ from this equation:
$$
10000x - x = 1331.\overline{1331} - 0.\overline{1331}
$$
$$
9999x = 1331
$$

---

### Step 3: Solve for $x$

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

---

### Step 4: Simplify the fraction

We simplify this fraction by finding the greatest common divisor (GCD) of 1331 and 9999.

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

So the GCD is $11$, which is the only common prime factor.

Divide numerator and denominator by 11:
$$
\frac{1331}{9999} = \frac{1331 \div 11}{9999 \div 11} = \frac{121}{909}
$$

Now, check whether 121 and 909 have any common factors. The prime factorization of:
- $121 = 11^2$
- $909 = 3^2 \times 101$

There are no common prime factors, so the fraction is already in lowest terms.

---

### Step 5: Find $a + b$

Here, the simplified fraction is $ \frac{121}{909} $, so:
$$
a = 121, \quad b = 909
$$
$$
a + b = 121 + 909 = \boxed{1030}
$$

[Assistant]
Explanation:
Okay, let's see. I need to convert the repeating decimal 0.133113311331... into a fraction. The question says it's a repeating decimal, right? So the bar is over 1331, which means the digits 1331 repeat indefinitely. So the decimal is 0.133113311331...

First, I remember that there's a method to convert repeating decimals to fractions. Let me recall how that works. I think you set the decimal equal to a variable, multiply both sides by a power of 10 to move the decimal point so that the repeating parts align, then subtract the equations to eliminate the repeating part. Let me try to apply that here.

Let me denote the repeating decimal as x. So let me write:

Let x = 0.133113311331...

Now, the repeating part is "1331", which is four digits. So the decimal repeats every four digits. Therefore, if I multiply x by 10^4 (which is 10000), I should shift the decimal point four places to the right. Let's try that:

10000x = 1331.133113311331...

But the original x is 0.133113311331... So if I subtract x from 10000x, maybe the repeating parts will cancel out.

So let me write:

10000x = 1331.133113311331...

minus

x = 0.133113311331...

Subtracting the two equations:

10000x - x = 1331.133113311331... - 0.133113311331...

Which gives:

9999x = 1331

So then, solving for x:

x = 1331 / 9999

Wait, but I need to check if this fraction can be simplified. 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.

Let me compute GCD(1331, 9999). First, note that 1331 is a number I recognize. 11^3 is 1331, because 11*11 is 121, 121*11 is 1331. Yes, that's right. So 1331 is 11³.

Now, let's factorize 9999. Let me see. 9999 is 10000 - 1, which factors into (100 - 1)(100 + 1) = 99*101. Wait, 99 is 9*11, and 101 is a prime number. Therefore:

9999 = 9 * 11 * 101.

So 9999 factors into 9 * 11 * 101.

So, 1331 is 11³, and 9999 is 9 * 11 * 101. Therefore, the GCD of 1331 and 9999 is the product of the common prime factors with the lowest exponents. The common prime factor is 11. The exponent of 11 in 1331 is 3, and in 9999 it's 1. So the GCD is 11^1 = 11.

Therefore, we can divide numerator and denominator by 11:

Numerator: 1331 / 11 = let me calculate that. 11*121 is 1331, right? Because 11*120=1320, plus 11 is 1331. So 1331 / 11 = 121.

Denominator: 9999 / 11 = let me compute that. 9999 divided by 11. Let's see, 11*900 = 9900. Then 9999 - 9900 = 99. 99 divided by 11 is 9. So total is 900 + 9 = 909. Therefore, 9999 / 11 = 909.

Therefore, the simplified fraction is 121 / 909.

Wait, but now I need to check if 121 and 909 have any common factors. Let's check GCD(121, 909).

121 is 11². Let's factorize 909. Let's divide 909 by 3: 9 + 0 + 9 = 18, which is divisible by 3. So 909 ÷ 3 = 303. Then 303 ÷ 3 = 101. Therefore, 909 is 3² * 101.

So, prime factors of 121 are 11², and prime factors of 909 are 3² * 101. There are no common factors between 121 and 909, since 11 is not a factor of 909 (since 909 ÷ 11 is 82.63... which isn't integer). Therefore, the fraction 121/909 is in simplest terms.

Therefore, the original decimal 0.13311331... is equal to 121/909, so a is 121 and b is 909. Therefore, a + b is 121 + 909 = 1030.

Wait, let me check again. Let me verify if the initial steps were correct.

Starting with x = 0.13311331..., multiplied by 10^4 = 10000 gives 1331.13311331...

Subtracting x gives 9999x = 1331. So x = 1331 / 9999. Then simplifying. Yes, that seems correct.

Then GCD(1331, 9999) is 11. Dividing numerator and denominator by 11 gives 121 / 909, which reduces to lowest terms. Then 121 + 909 is 1030.

But wait, let me check if 121 and 909 have any common factors. As I said before, 121 is 11². 909 divided by 11 is 82.636..., which is not integer. Let me check 11*82 = 902, 909 - 902 = 7. So remainder 7. Therefore, 11 is not a factor of 909. Therefore, there are no more common factors. Therefore, 121/909 is reduced.

But wait, let me check if 909 has factors of 11? 909 divided by 11 is 82.63... as above, so no. So the fraction is reduced.

Therefore, a + b is 121 + 909 = 1030. Wait, but let me check with another method to be sure.

Alternatively, maybe I made a mistake in the initial setup?

Let me check by converting 121/909 to decimal. Let me perform the division 121 ÷ 909.

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

Multiply numerator and denominator by something? Maybe not. Alternatively, since 1/909 is approximately... Well, perhaps it's easier to check if 121/909 equals 0.1331...

Alternatively, let me do the division 121 divided by 909.

First, note that 909 goes into 121 zero times. Add decimal point, 121.0000...

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 2727 from 3010: 3010 - 2727 = 283. Bring down a zero: 2830.

909 goes into 2830 three times again (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 is where we started after the first step. So the decimal repeats from here. So the decimal expansion is 0.1331331... So 0.1331331..., which is 0.1331 repeating? Wait, but the original decimal was 0.133113311331...

Wait, hold on. Did I get the decimal expansion right?

Wait, according to the division, 121/909 is 0.1331331331..., so the repeating part is "1331", but let me check:

Starting the division:

After decimal:

First step: 121.0000...

First digit: 1, remainder 301.

Second digit: 3, remainder 283.

Third digit: 3, remainder 103.

Fourth digit: 1, remainder 121.

Then it repeats. So the decimal is 0.1331 1331 1331..., so it's 0.133113311331..., which matches the original decimal given in the problem. Therefore, 121/909 is indeed equal to 0.13311331..., so the fraction is correct.

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

Wait, but let me check once more if the initial equation was correct. I set x = 0.13311331..., multiplied by 10000 to get 1331.1331..., then subtracted x to get 9999x = 1331. So x = 1331 / 9999. Then simplified.

But let me check that the decimal is indeed repeating every four digits. The original decimal is 0.133113311331... So the repeating part is "1331", which is four digits. So the method is correct because multiplying by 10^4 shifts the decimal four places, so that subtracting x cancels out the repeating part.

Therefore, all steps seem correct.

But let me check if 1331/9999 reduces to 121/909. 1331 divided by 11 is 121, and 9999 divided by 11 is 909. Then, as we saw, 121 and 909 have no common factors, so yes.

Therefore, the answer is 121 + 909 = 1030.

But wait, just to make sure, let me check if there is another way to think about this. Maybe I made a mistake in the GCD?

Let me recalculate GCD(1331, 9999). Using the Euclidean algorithm:

GCD(9999, 1331). So divide 9999 by 1331.

1331 * 7 = 9317. Subtract that from 9999: 9999 - 9317 = 682.

So GCD(1331, 682).

Now divide 1331 by 682. 682 * 1 = 682. Subtract from 1331: 1331 - 682 = 649.

So GCD(682, 649).

Divide 682 by 649: 649 * 1 = 649. Subtract from 682: 682 - 649 = 33.

GCD(649, 33).

Divide 649 by 33: 33*19 = 627. Subtract: 649 - 627 = 22.

GCD(33, 22).

Divide 33 by 22: 22*1 = 22. Subtract: 33 - 22 = 11.

GCD(22, 11).

Divide 22 by 11: 11*2 = 22. Remainder 0. So GCD is 11.

Therefore, GCD(1331, 9999) is indeed 11. Therefore, the simplification is correct.

So, yes, the answer is 121 + 909 = 1030. Therefore, I think that's correct.

But just to be absolutely thorough, let me check if 121 and 909 have any common factors. As we saw earlier, 121 is 11². Let's check if 11 divides 909. 9 + 0 + 9 = 18, which is not divisible by 11. Wait, the divisibility rule for 11 is alternating sum of digits. Let me apply that.

For 909: digits are 9, 0, 9. Alternating sum: 9 - 0 + 9 = 18. 18 is not divisible by 11, so 909 is not divisible by 11. Therefore, GCD(121, 909) = 1. So yes, 121/909 is reduced.

Therefore, the final answer is 121 + 909 = 1030. Wait, but let me check with another approach. Maybe the decimal is not four digits? Wait, the original decimal is 0.133113311331... So starting from the decimal, the first four digits after the decimal are 1331, then repeating. So the repeating block is four digits. So the method of multiplying by 10^4 is correct. Therefore, the setup was correct.

Alternatively, suppose someone thought that the repeating part is "331" or something else? But the problem says 0.\overline{1331}, which indicates that the bar is over 1331, so the repeating part is four digits: 1, 3, 3, 1. Therefore, the setup is correct.

Therefore, I think the answer is 1030. Therefore, I think that is the correct answer.

**Final Answer**
The value of $a + b$ is \boxed{1030}.

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

---

### Step 1: Let the decimal be represented by a variable

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

---

### Step 2: Multiply by a power of 10 to align the repeating block

Since the repeating block "1331" has 4 digits, we multiply both sides by $10^4 = 10000$ to move the decimal point four places to the right:
$$
10000x = 1331.\overline{1331}
$$

Now subtract the original $x$ from this equation:
$$
10000x - x = 1331.\overline{1331} - 0.\overline{1331}
$$
$$
9999x = 1331
$$

---

### Step 3: Solve for $x$

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

---

### Step 4: Simplify the fraction

We simplify this fraction by finding the greatest common divisor (GCD) of 1331 and 9999.

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

So the GCD is $11$, which is the only common prime factor.

Divide numerator and denominator by 11:
$$
\frac{1331}{9999} = \frac{1331 \div 11}{9999 \div 11} = \frac{121}{909}
$$

Now, check whether 121 and 909 have any common factors. The prime factorization of:
- $121 = 11^2$
- $909 = 3^2 \times 101$

There are no common prime factors, so the fraction is already in lowest terms.

---

### Step 5: Find $a + b$

Here, the simplified fraction is $ \frac{121}{909} $, so:
$$
a = 121, \quad b = 909
$$
$$
a + b = 121 + 909 = \boxed{1030}
$$

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 cost 1200 lire in Italy and $1.50 in the U.S. So, they want me to find the exchange rate between lire and dollars using this information. Then, using that exchange rate, convert 1,000,000 lire to dollars.

First, let me recall what an exchange rate is. The exchange rate tells you how much of one currency you can get for a unit of another currency. For example, if the exchange rate is 1 dollar = 1000 lire, then 1 lire is 0.001 dollars.

So, here, we have two prices for the same product: 1200 lire in Italy and $1.50 in the U.S. So, the exchange rate should relate these two amounts. Let me think. If I want to find how much 1200 lire is in dollars, it should be equal to $1.50. So, that gives me a ratio between lire and dollars.

Let me write that down as an equation. Let me denote the exchange rate as X lire per dollar. Wait, but I need to be careful here. Maybe I should think in terms of how many dollars one lire is worth, or how many lire one dollar is worth.

Alternatively, since 1200 lire equals $1.50, I can find out how many lire equal one dollar, or how many dollars equal one lire.

Let me try both ways.

First, let's find out how many lire equal one dollar.

If 1200 lire = $1.50, then to find how many lire equal $1, I can divide both sides by 1.50.

So, 1200 lire / 1.50 dollars = 800 lire per dollar.

Wait, that would be 1200 divided by 1.5. Let me compute that.

1200 divided by 1.5. Let me think. 1.5 goes into 1200 how many times? Well, 1.5 times 800 is 1200. So, yes, 800 lire per dollar. So, 1 dollar equals 800 lire.

Alternatively, if I wanted to find how many dollars equal 1 lire, I can take the reciprocal. So, 1 lire equals 1/800 dollars.

So, now, if I want to convert 1,000,000 lire to dollars, I can multiply by the exchange rate of 1/800 dollars per lire.

So, 1,000,000 lire * (1 dollar / 800 lire) = 1,000,000 / 800 dollars.

Let me compute that.

First, let's simplify 1,000,000 divided by 800.

Well, 1,000,000 divided by 800. Let me divide numerator and denominator by 100 first to make it simpler: 10,000 / 8.

Wait, 1,000,000 divided by 100 is 10,000. 800 divided by 100 is 8.

Yes, so 10,000 divided by 8.

10,000 divided by 8 is 1250.

Wait, 8 times 1250 is 10,000? Let me check.

8 * 1000 = 8000.

8 * 250 = 2000.

So 8000 + 2000 = 10,000. Yes, that's correct.

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

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

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

Original information: 1200 lire = $1.50.

So, exchange rate: 1200 lire = 1.5 dollars.

So, to find how many dollars is 1 lire, divide both sides by 1200:

1 lire = 1.5 dollars / 1200.

Which is 1.5 / 1200 dollars per lire.

Calculating that: 1.5 divided by 1200.

Well, 1.5 divided by 1200. Let me compute that.

First, 1.5 / 1200 = (3/2) / 1200 = 3 / 2400 = 1 / 800.

So, yes, 1 lire equals 1/800 dollars. Which is 0.00125 dollars.

Therefore, 1,000,000 lire would be 1,000,000 * (1/800) = 1,000,000 / 800 = 1250 dollars.

Yes, same result.

Alternatively, another way to think about it is if 1200 lire equals $1.50, then how much is 1,000,000 lire?

We can set up a proportion.

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

So, cross-multiplying:

1200 * x = 1.50 * 1,000,000

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

Compute that:

First, 1.5 * 1,000,000 = 1,500,000

Then divide by 1200: 1,500,000 / 1200

Simplify: divide numerator and denominator by 100: 15,000 / 12

Divide numerator and denominator by 12:

Wait, 15,000 divided by 12. Let me do this division.

12 * 1250 = 15,000.

Yes, because 12 * 1000 = 12,000; 12 * 250 = 3,000; so total 15,000.

Therefore, x = 1250.

Same answer.

Therefore, regardless of the method, the answer seems to be 1250 dollars.

Let me check again with another approach.

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

As above, 1 lire = 1.50 / 1200 dollars.

Which is 0.00125 dollars per lire.

So, multiplying that by 1,000,000 lire:

0.00125 * 1,000,000 = 1250 dollars.

Same result.

So, all approaches lead to $1250. Therefore, I think that is the correct answer.

But just to make sure I didn't mix up the exchange rate.

Wait, sometimes exchange rates can be quoted as dollars per lire or lire per dollar. So, let me confirm that.

If the exchange rate is 800 lire per dollar, that means 1 dollar = 800 lire. So, to get dollars from lire, you divide by 800.

Alternatively, if the exchange rate is 0.00125 dollars per lire, which is 1/800, so again, multiplying by that gives the dollar amount.

So, for example, 1200 lire * (1 dollar / 800 lire) = 1.5 dollars. Which matches the given information.

Therefore, yes, the exchange rate is correct.

Therefore, converting 1,000,000 lire:

1,000,000 / 800 = 1250.

So, 1250 dollars.

I think that's solid.

Alternatively, if I wanted to think of it as how many dollars per lire, which is 1.50 / 1200.

Which is 0.00125 dollars per lire.

So, 1,000,000 * 0.00125 = 1,000,000 * 1.25 * 10^(-3) = 1,000,000 * 0.00125.

Calculating 1,000,000 * 0.00125.

Well, 1,000,000 * 0.001 is 1,000.

Then 1,000,000 * 0.00025 is 250.

So total is 1,000 + 250 = 1,250.

Same answer again.

So, I think I've checked this multiple ways.

Therefore, I can be confident that 1,000,000 lire is equivalent to $1250.

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

To determine how many dollars are equivalent to 1,000,000 lire in 1992, we begin by finding the exchange rate between the Italian lira and the United States dollar using the provided data:

> A scoop of gelato cost **1200 lire** in Italy and **$1.50** in the U.S.

---

### **Step 1: Establish the Exchange Rate**

We’re told that 1200 lire is equivalent to $1.50. This gives us a key ratio:

$$
1200 \text{ lire} = \$1.50
$$

To find how much **1 lire** is worth in dollars, divide both sides of the equation by 1200:

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

$$
1 \text{ lire} = 0.00125 \text{ dollars}
$$

This means **1 lire = $0.00125**.

Alternatively, we can compute how many **lire** are in **1 dollar**, by dividing both sides by $1.50:

$$
1 \text{ dollar} = \frac{1200}{1.50} \text{ lire}
$$

$$
1 \text{ dollar} = 800 \text{ lire}
$$

So the exchange rate can equally be expressed as:

$$
1 \text{ dollar} = 800 \text{ lire} \quad \text{or} \quad 1 \text{ lire} = \frac{1}{800} \text{ dollars}
$$

---

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

Now that we know the value of 1 lire in dollars, we can compute the value of 1,000,000 lire as follows:

$$
1,000,000 \text{ lire} \times \frac{1}{800} \text{ dollars/lire} = \frac{1,000,000}{800} \text{ dollars}
$$

$$
\frac{1,000,000}{800} = 1250 \text{ dollars}
$$

---

### **Conclusion**

Using the given exchange rate and applying basic unit conversion techniques, we find that:

$$
\boxed{1250}
$$

dollars are equivalent to 1,000,000 lire in 1992.

[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 cost 1200 lire in Italy and $1.50 in the U.S. So, they want me to find the exchange rate between lire and dollars using this information. Then, using that exchange rate, convert 1,000,000 lire to dollars.

First, let me recall what an exchange rate is. The exchange rate tells you how much of one currency you can get for a unit of another currency. For example, if the exchange rate is 1 dollar = 1000 lire, then 1 lire is 0.001 dollars.

So, here, we have two prices for the same product: 1200 lire in Italy and $1.50 in the U.S. So, the exchange rate should relate these two amounts. Let me think. If I want to find how much 1200 lire is in dollars, it should be equal to $1.50. So, that gives me a ratio between lire and dollars.

Let me write that down as an equation. Let me denote the exchange rate as X lire per dollar. Wait, but I need to be careful here. Maybe I should think in terms of how many dollars one lire is worth, or how many lire one dollar is worth.

Alternatively, since 1200 lire equals $1.50, I can find out how many lire equal one dollar, or how many dollars equal one lire.

Let me try both ways.

First, let's find out how many lire equal one dollar.

If 1200 lire = $1.50, then to find how many lire equal $1, I can divide both sides by 1.50.

So, 1200 lire / 1.50 dollars = 800 lire per dollar.

Wait, that would be 1200 divided by 1.5. Let me compute that.

1200 divided by 1.5. Let me think. 1.5 goes into 1200 how many times? Well, 1.5 times 800 is 1200. So, yes, 800 lire per dollar. So, 1 dollar equals 800 lire.

Alternatively, if I wanted to find how many dollars equal 1 lire, I can take the reciprocal. So, 1 lire equals 1/800 dollars.

So, now, if I want to convert 1,000,000 lire to dollars, I can multiply by the exchange rate of 1/800 dollars per lire.

So, 1,000,000 lire * (1 dollar / 800 lire) = 1,000,000 / 800 dollars.

Let me compute that.

First, let's simplify 1,000,000 divided by 800.

Well, 1,000,000 divided by 800. Let me divide numerator and denominator by 100 first to make it simpler: 10,000 / 8.

Wait, 1,000,000 divided by 100 is 10,000. 800 divided by 100 is 8.

Yes, so 10,000 divided by 8.

10,000 divided by 8 is 1250.

Wait, 8 times 1250 is 10,000? Let me check.

8 * 1000 = 8000.

8 * 250 = 2000.

So 8000 + 2000 = 10,000. Yes, that's correct.

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

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

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

Original information: 1200 lire = $1.50.

So, exchange rate: 1200 lire = 1.5 dollars.

So, to find how many dollars is 1 lire, divide both sides by 1200:

1 lire = 1.5 dollars / 1200.

Which is 1.5 / 1200 dollars per lire.

Calculating that: 1.5 divided by 1200.

Well, 1.5 divided by 1200. Let me compute that.

First, 1.5 / 1200 = (3/2) / 1200 = 3 / 2400 = 1 / 800.

So, yes, 1 lire equals 1/800 dollars. Which is 0.00125 dollars.

Therefore, 1,000,000 lire would be 1,000,000 * (1/800) = 1,000,000 / 800 = 1250 dollars.

Yes, same result.

Alternatively, another way to think about it is if 1200 lire equals $1.50, then how much is 1,000,000 lire?

We can set up a proportion.

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

So, cross-multiplying:

1200 * x = 1.50 * 1,000,000

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

Compute that:

First, 1.5 * 1,000,000 = 1,500,000

Then divide by 1200: 1,500,000 / 1200

Simplify: divide numerator and denominator by 100: 15,000 / 12

Divide numerator and denominator by 12:

Wait, 15,000 divided by 12. Let me do this division.

12 * 1250 = 15,000.

Yes, because 12 * 1000 = 12,000; 12 * 250 = 3,000; so total 15,000.

Therefore, x = 1250.

Same answer.

Therefore, regardless of the method, the answer seems to be 1250 dollars.

Let me check again with another approach.

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

As above, 1 lire = 1.50 / 1200 dollars.

Which is 0.00125 dollars per lire.

So, multiplying that by 1,000,000 lire:

0.00125 * 1,000,000 = 1250 dollars.

Same result.

So, all approaches lead to $1250. Therefore, I think that is the correct answer.

But just to make sure I didn't mix up the exchange rate.

Wait, sometimes exchange rates can be quoted as dollars per lire or lire per dollar. So, let me confirm that.

If the exchange rate is 800 lire per dollar, that means 1 dollar = 800 lire. So, to get dollars from lire, you divide by 800.

Alternatively, if the exchange rate is 0.00125 dollars per lire, which is 1/800, so again, multiplying by that gives the dollar amount.

So, for example, 1200 lire * (1 dollar / 800 lire) = 1.5 dollars. Which matches the given information.

Therefore, yes, the exchange rate is correct.

Therefore, converting 1,000,000 lire:

1,000,000 / 800 = 1250.

So, 1250 dollars.

I think that's solid.

Alternatively, if I wanted to think of it as how many dollars per lire, which is 1.50 / 1200.

Which is 0.00125 dollars per lire.

So, 1,000,000 * 0.00125 = 1,000,000 * 1.25 * 10^(-3) = 1,000,000 * 0.00125.

Calculating 1,000,000 * 0.00125.

Well, 1,000,000 * 0.001 is 1,000.

Then 1,000,000 * 0.00025 is 250.

So total is 1,000 + 250 = 1,250.

Same answer again.

So, I think I've checked this multiple ways.

Therefore, I can be confident that 1,000,000 lire is equivalent to $1250.

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

To determine how many dollars are equivalent to 1,000,000 lire in 1992, we begin by finding the exchange rate between the Italian lira and the United States dollar using the provided data:

> A scoop of gelato cost **1200 lire** in Italy and **$1.50** in the U.S.

---

### **Step 1: Establish the Exchange Rate**

We’re told that 1200 lire is equivalent to $1.50. This gives us a key ratio:

$$
1200 \text{ lire} = \$1.50
$$

To find how much **1 lire** is worth in dollars, divide both sides of the equation by 1200:

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

$$
1 \text{ lire} = 0.00125 \text{ dollars}
$$

This means **1 lire = $0.00125**.

Alternatively, we can compute how many **lire** are in **1 dollar**, by dividing both sides by $1.50:

$$
1 \text{ dollar} = \frac{1200}{1.50} \text{ lire}
$$

$$
1 \text{ dollar} = 800 \text{ lire}
$$

So the exchange rate can equally be expressed as:

$$
1 \text{ dollar} = 800 \text{ lire} \quad \text{or} \quad 1 \text{ lire} = \frac{1}{800} \text{ dollars}
$$

---

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

Now that we know the value of 1 lire in dollars, we can compute the value of 1,000,000 lire as follows:

$$
1,000,000 \text{ lire} \times \frac{1}{800} \text{ dollars/lire} = \frac{1,000,000}{800} \text{ dollars}
$$

$$
\frac{1,000,000}{800} = 1250 \text{ dollars}
$$

---

### **Conclusion**

Using the given exchange rate and applying basic unit conversion techniques, we find that:

$$
\boxed{1250}
$$

dollars are equivalent to 1,000,000 lire in 1992.

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, let me think about the factors of 53. First, I know that 53 is a prime number. Wait, is that right? Let me check. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. Let me try dividing 53 by some numbers.

Starting with 2: 53 divided by 2 is 26.5, which isn't an integer. Divided by 3? 3 times 17 is 51, and 53 minus 51 is 2, so no. Divided by 5? Ends with 3, so no. Divided by 7? 7*7 is 49, 7*8 is 56, which is too big. 53-49=4, so not divisible by 7. 11? 11*4 is 44, 11*5 is 55, too big. So nope. So 53 is a prime number.

If 53 is prime, then its only factors are 1 and itself. Which means that the square root of 53 cannot be simplified further because there are no perfect square factors other than 1. Therefore, √53 is already in its simplest radical form.

Wait, but just to be thorough, let me check if I made a mistake in determining if 53 is prime. Maybe there's a factor I missed? Let me list all the primes less than the square root of 53. The square root of 53 is approximately 7.28, so primes less than or equal to 7 are 2, 3, 5, 7. I already checked those. So no factors there. So yes, 53 is prime.

Therefore, √53 cannot be simplified. So the answer should just be √53.

But let me also think about how to check if a number is a perfect square. For example, if I think of numbers like 49 (7²), 64 (8²), etc. 53 is between 49 and 64, so it's not a perfect square. Therefore, its square root is irrational. So, it can't be simplified into a whole number or a simpler radical expression.

Alternatively, maybe I can try to write 53 as a product of two numbers where one is a square? For example, suppose I try 53 = 1 * 53. Well, 1 is a perfect square, but that doesn't help because √(1*53) = √1 * √53 = 1*√53, which is just √53. So that doesn't simplify anything.

Alternatively, if there was another factor pair, say 53 = something else. But since it's prime, there are no other factor pairs except 1 and 53. Therefore, there's no way to write it as a product of a square number and another number except 1*53. Hence, it can't be simplified.

So, I think I can confirm that √53 is already in its simplest radical form.

But just to double-check, let me think of another approach. Suppose I try to perform prime factorization on 53. Since 53 is prime, its prime factorization is just 53. So, when you take the square root of that, you can't break it down further.

Another way to think about it: if there was a square factor, then that square factor would have to be a square number, which is a product of primes raised to even exponents. Since 53 only has one prime factor, 53, raised to the first power, there are no even exponents here. Therefore, the square root cannot be simplified.

Yeah, that makes sense. So, all paths lead to the conclusion that √53 is already in simplest radical form.

Wait, but maybe I should check with an example. For instance, √50 can be simplified because 50 = 25*2, and 25 is a perfect square. So √50 = √25*√2 = 5√2. But 53 doesn't have any such factors, so no simplification is possible.

Alternatively, let me think about numbers close to 53. For example, 49 is 7 squared, 64 is 8 squared. 53 is between them, so the square root is between 7 and 8. If I wanted to approximate it, maybe 7.28 or something, but that's not necessary here. The question is about simplest radical form, not decimal approximation.

Another thought: sometimes people might confuse whether numbers like 53 can be broken down into a product of square numbers and other numbers. But since 53 is prime, and square numbers have even exponents in their prime factors, there's no way to write 53 as such a product.

So, after all these checks, I think I can confidently say that √53 is already in its simplest radical form. Therefore, the answer is √53.

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

To determine the simplest radical form of $ \sqrt{53} $, we begin by analyzing whether 53 can be factored into a product that includes a perfect square, which would allow simplification.

---

### Step 1: Determine if 53 is a Prime Number

A number is prime if it has no divisors other than 1 and itself. To check if 53 is prime, we attempt to divide it by all prime numbers less than or equal to $ \sqrt{53} \approx 7.28 $:

- 53 is not divisible by 2 (odd number)
- Not divisible by 3: $ 5 + 3 = 8 $, not divisible by 3
- Not divisible by 5: ends in 3
- Not divisible by 7: $ 7 \times 7 = 49 $, $ 7 \times 8 = 56 $, so not divisible by 7

Thus, 53 has no factors other than 1 and itself, confirming it is a **prime number**.

---

### Step 2: Examine the Square Root

Since 53 is prime, its only factor pair is $ 1 \times 53 $.

For $ \sqrt{53} $ to be simplified, the radicand must contain a factor that is a perfect square (i.e., a product of primes with even exponents). However, $ 53 $ is a single prime factor raised to the power of 1, meaning there are **no even exponents**, and thus **no perfect square factors**.

This confirms that $ \sqrt{53} $ **cannot be simplified** further using radicals.

---

### Step 3: Conclusion

As 53 has no perfect square factors other than 1, and its prime factorization does not allow for simplification, we conclude that:

$$
\sqrt{53}
$$

is already in **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, let me think about the factors of 53. First, I know that 53 is a prime number. Wait, is that right? Let me check. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. Let me try dividing 53 by some numbers.

Starting with 2: 53 divided by 2 is 26.5, which isn't an integer. Divided by 3? 3 times 17 is 51, and 53 minus 51 is 2, so no. Divided by 5? Ends with 3, so no. Divided by 7? 7*7 is 49, 7*8 is 56, which is too big. 53-49=4, so not divisible by 7. 11? 11*4 is 44, 11*5 is 55, too big. So nope. So 53 is a prime number.

If 53 is prime, then its only factors are 1 and itself. Which means that the square root of 53 cannot be simplified further because there are no perfect square factors other than 1. Therefore, √53 is already in its simplest radical form.

Wait, but just to be thorough, let me check if I made a mistake in determining if 53 is prime. Maybe there's a factor I missed? Let me list all the primes less than the square root of 53. The square root of 53 is approximately 7.28, so primes less than or equal to 7 are 2, 3, 5, 7. I already checked those. So no factors there. So yes, 53 is prime.

Therefore, √53 cannot be simplified. So the answer should just be √53.

But let me also think about how to check if a number is a perfect square. For example, if I think of numbers like 49 (7²), 64 (8²), etc. 53 is between 49 and 64, so it's not a perfect square. Therefore, its square root is irrational. So, it can't be simplified into a whole number or a simpler radical expression.

Alternatively, maybe I can try to write 53 as a product of two numbers where one is a square? For example, suppose I try 53 = 1 * 53. Well, 1 is a perfect square, but that doesn't help because √(1*53) = √1 * √53 = 1*√53, which is just √53. So that doesn't simplify anything.

Alternatively, if there was another factor pair, say 53 = something else. But since it's prime, there are no other factor pairs except 1 and 53. Therefore, there's no way to write it as a product of a square number and another number except 1*53. Hence, it can't be simplified.

So, I think I can confirm that √53 is already in its simplest radical form.

But just to double-check, let me think of another approach. Suppose I try to perform prime factorization on 53. Since 53 is prime, its prime factorization is just 53. So, when you take the square root of that, you can't break it down further.

Another way to think about it: if there was a square factor, then that square factor would have to be a square number, which is a product of primes raised to even exponents. Since 53 only has one prime factor, 53, raised to the first power, there are no even exponents here. Therefore, the square root cannot be simplified.

Yeah, that makes sense. So, all paths lead to the conclusion that √53 is already in simplest radical form.

Wait, but maybe I should check with an example. For instance, √50 can be simplified because 50 = 25*2, and 25 is a perfect square. So √50 = √25*√2 = 5√2. But 53 doesn't have any such factors, so no simplification is possible.

Alternatively, let me think about numbers close to 53. For example, 49 is 7 squared, 64 is 8 squared. 53 is between them, so the square root is between 7 and 8. If I wanted to approximate it, maybe 7.28 or something, but that's not necessary here. The question is about simplest radical form, not decimal approximation.

Another thought: sometimes people might confuse whether numbers like 53 can be broken down into a product of square numbers and other numbers. But since 53 is prime, and square numbers have even exponents in their prime factors, there's no way to write 53 as such a product.

So, after all these checks, I think I can confidently say that √53 is already in its simplest radical form. Therefore, the answer is √53.

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

To determine the simplest radical form of $ \sqrt{53} $, we begin by analyzing whether 53 can be factored into a product that includes a perfect square, which would allow simplification.

---

### Step 1: Determine if 53 is a Prime Number

A number is prime if it has no divisors other than 1 and itself. To check if 53 is prime, we attempt to divide it by all prime numbers less than or equal to $ \sqrt{53} \approx 7.28 $:

- 53 is not divisible by 2 (odd number)
- Not divisible by 3: $ 5 + 3 = 8 $, not divisible by 3
- Not divisible by 5: ends in 3
- Not divisible by 7: $ 7 \times 7 = 49 $, $ 7 \times 8 = 56 $, so not divisible by 7

Thus, 53 has no factors other than 1 and itself, confirming it is a **prime number**.

---

### Step 2: Examine the Square Root

Since 53 is prime, its only factor pair is $ 1 \times 53 $.

For $ \sqrt{53} $ to be simplified, the radicand must contain a factor that is a perfect square (i.e., a product of primes with even exponents). However, $ 53 $ is a single prime factor raised to the power of 1, meaning there are **no even exponents**, and thus **no perfect square factors**.

This confirms that $ \sqrt{53} $ **cannot be simplified** further using radicals.

---

### Step 3: Conclusion

As 53 has no perfect square factors other than 1, and its prime factorization does not allow for simplification, we conclude that:

$$
\sqrt{53}
$$

is already in **simplest radical form**.

---

### Final Answer

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

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