mathschallenge.net

Problem

Consider the following results.

8¹1 = 7 = 71
8²1 = 63 = 79
8³1 = 511 = 773
8⁴1 = 4095 = 7585
8⁵1 = 32767 = 74681

Prove that 8ⁿ1 is always divisible by 7.

Solution

Clearly it is true for n=1: 8¹1 = 7.

Assume that it is true for n=k: 8^k1 is divisible by 7.

Consider the next case, n=k+1.

8^k+11 = 8^k81 = 8(8^k1)+81 = 8(8^k1)+7.

That is, if 8^k1 is divisible by 7, 8^k+11 will also divide evenly by 8. As it works for n=1, it must be true for all n.

Prove that aⁿ1 is always divisible by a1.