12th CanMO 1980 Problem 1 If the 5-digit decimal number a679b is a multiple of 72 find a and b.
100 = 28 mod 72, 1000 = -8 mod 72, 10000 = -8 mod 72, so we require -8a - 48 + 196 + 90 + b = 0 mod 72, or 22 - 8a + b = 0 mod 72. We have -72 < -50 = 22 - 8.9 + 0 ≤ 22 - 8a + b ≤ 22 + 0 + 9 = 31 < 72, so 22 - 8a + b = 0, or 8a = 22 + b. So b = 2, a = 3. Check: 36792 = 511·72. 12th CanMO 1980 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002