The Last Digit

A problem from the 1996 Georg Mohr mathematics competition in Denmark:

n is a positive integer. The next-to-last digit in the decimal expression of n² is 7. What’s the last digit?

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This solution is by Michel Bataille of Rouen, France: Write n as 10a + b, where a is a nonnegative integer and b is a single nonnegative digit. Then n² = 100a² + 20ab + b², and the last two digits of n² are the same as those of 20ab + b². Now, 20ab is 10 × 2ab — that is, it’s an even number multiplied by 10. So it must end with 00, 20, 40, 60, or 80. We know that the next-to-last digit of 20ab + b² is 7, so the next-to-last digit of b² must be odd. That can occur only when b is 4 or 6, giving b² = 16 or 36. In both cases the last digit is 6. (“The Olympiad Corner,” Crux Mathematicorum 27:4 [May 2001], 239.)