What is the remainder when 7^{n}-6n is divided by 36 for n=100?

Explanation

We can express 7^n = (1+6)^n. Using binomial expansion, (1+6)^n = 1 + 6n + \binom{n}{2}6^2 + \dots + 6^n. Rearranging the terms gives 7^n - 6n = 1 + 36[\binom{n}{2} + \dots], which means 7^n - 6n = 1 + 36k for some integer k. Thus, dividing by 36 always leaves a remainder of 1.

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