Question

10^n-2^n divided by 5^n-2^n

Answer 1

Answer:

n5−nn5−n is a multiple of 5 ⇔⇔ n5+10n4+35n3+50n2+24n=n5−n+5(2n4+7n3+10n2+5n)n5+10n4+35n3+50n2+24n=n5−n+5(2n4+7n3+10n2+5n) is a multiple of 5. The latter is just n(n+1)(n+2)(n+3)(n+4)n(n+1)(n+2)(n+3)(n+4), which is the product of 5 consecutive integers, hence is a multiple of 5. Note: You should generally be able to do the above transformation, and can take the product of any 5 (or k) consecutive integers, if you are looking at a polynomial of degree 5 (or k).

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