The Problem:
We are given a four-digit number that leaves a remainder of 29 when divided by 65. We need to find the remainder when the same number is divided by 13.

Understanding the Remainder:
When we divide a number by another number, the remainder is the amount left over after the division process. For example, when we divide 10 by 3, we get a quotient of 3 and a remainder of 1.

Divisibility Rule:
To find the remainder when a number is divided by another number, we can use the concept of divisibility rules. The divisibility rule for 13 states that if the difference between twice the last digit of a number and the remaining digits is divisible by 13, then the number itself is divisible by 13.

Approach:
To find the remainder when the four-digit number is divided by 13, we can use the divisibility rule for 13. We need to check if the difference between twice the last digit and the remaining digits is divisible by 13.

Step-by-Step Solution:
1. Let's assume the four-digit number is represented as ABCD, where A, B, C, and D are the thousands, hundreds, tens, and units digits respectively.
2. We know that the number leaves a remainder of 29 when divided by 65. This can be expressed as:
ABCD ≡ 29 (mod 65)
3. To simplify the equation, we can rewrite it as:
1000A + 100B + 10C + D ≡ 29 (mod 65)
4. Simplifying further, we get:
35A + 4B + 6C + D ≡ 29 (mod 65)
5. Now, let's focus on the divisibility rule for 13. According to the rule, we need to check if the difference between twice the last digit (2D) and the remaining digits (35A + 4B + 6C) is divisible by 13.
6. So, we have:
2D - (35A + 4B + 6C) ≡ 0 (mod 13)
7. Rearranging the terms, we get:
2D - 35A - 4B - 6C ≡ 0 (mod 13)
8. Simplifying further, we get:
2D - 9A - 4B - 6C ≡ 0 (mod 13)
9. Comparing this equation with the given equation (step 4), we can see that the two equations are the same, except for the remainder values.
10. Therefore, the remainder when the four-digit number is divided by 13 is the same as the remainder when it is divided by 65, which is 29.
11. However, since the question asks for the remainder when divided by 13, the answer is 29 % 13 = 3.

Conclusion:
The remainder when the four-digit number is divided by 13 is 3. This is determined by applying the divisibility rule for 13 and simplifying the equation derived from the given information.