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True-False-Prove-Salvage. Let a ∈ N (so a is a constant). Claim: a|((a +1)^n −1) for n...

True-False-Prove-Salvage. Let a ∈ N (so a is a constant). Claim: a|((a +1)^n −1) for n ≥ 0.

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Answer #1

Smart proof using congruence :

We have, a 0 (mod a). (Since, a divides a)

So, a + 1 1 (mod a). (Adding 1 both sides)

So, (a+1)n 1n (mod n). (Raising to power n)

So, n | (a+1)n - 1 , for all n 0

Another proof using binomial theorem :

Expand (a+1)n using binomial theorem, & you will get the 1 cancelled out by (-1) at the end. Now take common a from all the remaining n terms, so, a divides (a+1)n - 1.

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