For a positive integer n , 25n−52n is divisible by:

Concept

For any positive integer n, the expression an − bn is always divisible by (a − b). This is a standard factor result for powers: (a − b) is a factor of an − bn for every positive integer n.

Application

1. Rewrite each power with a common exponent n. Since 25n = (25)n = 32n, and 52n = (52)n = 25n.

2. So the expression becomes 32n − 25n, which has the form an − bn with a = 32 and b = 25.

3. By the identity, (a − b) divides the expression. Here a − b = 32 − 25 = 7, so 7 divides 32n − 25n for every positive integer n.

Cross-check

Test n = 1: 32 − 25 = 7, divisible by 7. Test n = 2: 1024 − 625 = 399 = 7 × 57, divisible by 7. The divisibility by 7 holds in both, confirming the answer.