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

Understanding the Remainder:
When a number is divided by another number, the remainder is the amount left over after dividing the number as much as possible by the divisor. For example, when 10 is divided by 3, the quotient is 3 and the remainder is 1.

Divisibility Rule:
To solve this problem, we need to understand a key property of remainders and divisibility. If a number is divisible by both 65 and 13, it is also divisible by their least common multiple, which is 65. In other words, if a number leaves a remainder of 0 when divided by 65, it will also leave a remainder of 0 when divided by 13.

Dividing the Four-Digit Number by 65:
Let's assume the four-digit number is represented as ABCD, where A, B, C, and D are individual digits. When this number is divided by 65, the remainder is given as 29. Mathematically, we can represent this as:

ABCD = 65x + 29

Here, x represents the quotient when ABCD is divided by 65.

Finding the Remainder when Divided by 13:
To find the remainder when ABCD is divided by 13, we need to check if ABCD is divisible by 13. If it is, the remainder will be 0. If it is not divisible, we need to find the remainder.

Using the divisibility rule mentioned earlier, if ABCD is divisible by 65, it is also divisible by 13. Therefore, the remainder when ABCD is divided by 13 will be the same as the remainder when 29 is divided by 13.

Calculating the Remainder:
When we divide 29 by 13, the quotient is 2 and the remainder is 3. Therefore, the remainder when the four-digit number is divided by 13 is 3.

Final Answer:
The remainder when the four-digit number is divided by 13 is 3.