Proof of even like powers?

Question

Can someone show me the proof that difference of like even powers of any two numbers is divisible by the sum of the bases?

Answer 1

HINT: $(a^{2n} - b^{2n}) = (a^n - b^n)(a^n + b^n) =...$

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Added:

In response to a comment, this proof does not use logarithms.

The expression $a^n + b^n$ is called "the sum of nth powers". Notice that "the sum of the bases" is just $(a + b)$, which is the first factor in the factorization of the sum of nth powers.

Answer 2

Alternate Hint: $a \equiv-b \pmod{a+b}$