24th Eötvös 1917 Problem 2 A square has 10s digit 7. What is its units digit?
Suppose n = 100m + 10a + b. Then n 2 = 20ab + b 2 mod 100. So we are only interested in the last two digits of n. Also we require that b2 has an odd carry. Checking 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 we see that only 16 and 36 have an odd carry, and in both cases they give units digit 6. So the units digit must be 6. Note that there are such squares, eg 24 2 = 576. 24th Eötvös 1917 © John Scholes jscholes@kalva.demon.co.uk 1 Nov 2003 Last corrected/updated 1 Nov 03