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Mathematical Induction

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Find a prime number p such that 10^2n - p^2 is divisible by 3 for all positive integers n , and justify your choice by using mathematical induction.

最佳解答:

10^2n - p^2 = (10^n - p)(10^n + p) choose p = 2 (10^n + p) is divisible by 3 for all positive integers n because sum of all digits of (10^n + p) is divisible by 3 justify 10^2n - 2^2 is divisible by 3 for all positive integers n when n = 1, 10^2n - 2^2 = 96 is divisible by 3 suppose 10^2k - 4 is divisible by 3 for some positive integer k 10^2k - 4 = 3m, m is integer when n = k+1 10^2(k+1) - 4 = 100 (10^2k) - 4 = 100 (3m + 4) - 4 = 300m + 396 = 3(100m + 132) is divisible by 3 because 100m + 132 is integer ok...

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