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E26. Final digit of a power

The following is adapted from Posamentier and Salkind's Challenging Problems in Algebra, published by Dover Books.

(a) Prove that for any positive integer n, the final digit of n5 is the same as the final digit of n.
(b) Without using a computing device, find all positive integers whose 5-th power is a 7-digit number that ends in the digit 7.

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