When a number is divided by 13, the remainder is 10. When the same number is divided by 17, the remainder is 8. What is the number?

To solve this problem, we can use the Chinese Remainder Theorem. Here are the steps:

1. First, we need to find a number that when divided by 13 gives a remainder of 10 and when divided by 17 gives a remainder of 8. Let's denote this number as X.

2. According to the Chinese Remainder Theorem, we can express X as follows: X = a13 + 10 = b17 + 8, where a and b are some integers.

3. To find the smallest positive X, we can start by setting a = 1 and b = 1 and then gradually increase these values.

4. If we set a = 1, we get X = 13 + 10 = 23. But 23 divided by 17 does not give a remainder of 8.

5. If we set a = 2, we get X = 26 + 10 = 36. But 36 divided by 17 also does not give a remainder of 8.

6. We continue this process until we find a value of a that makes X divisible by 17 with a remainder of 8.

7. If we set a = 11, we get X = 143. And 143 divided by 17 gives a remainder of 8.

8. Therefore, the smallest positive number that satisfies the given conditions is 143.