Show using mathematical induction that 8^n - 1 is divisible by 7 for n=1,2,3,...
First step: n=1 we have 81 -1=7 which is divisible by 7. Assumption step: 8k-1 is divisible by 7. Induction step: Using the previous step we have that 8k-1=7x. So 8k = 7x+1. Therefore, 8k+1- 1=8(8k)-1=8(7x+1)-1 = 56x + 8 -1 = 56x+7 = 7(8x+1) which is divisible by 7. Hence, since it is true for n=1, n = k and for n=k+1 then it is true for all positive integers
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