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.