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A12. Fifth powers final digit - generalized.

If numbers are expressed in base b, for which b is it true that n5 and n end in the same digit for all positive integers n?
This is an obvious generalization of Problem E32.

E32. Fifth powers final digit

The following rather neat problem occurs in Challenging Problems in Algebra by Alfred S. Posamentier and Charles T. Salkind. I think it is suitable for a bright high school student or, with some hints, even for average high school students.

Prove that n and n5 always end in the same digit (in ordinary base-10 representation).