19th CanMO 1987 Problem 2 Find all the ways in which the number 1987 can be written in another base as a three digit number with the digits having the same sum 25.
45 2 = 2025. So any three digit number in base 45 or higher is too big. 12 3 = 1728, so any three digit number in base 12 or less is too small. Suppose the base is b and the number has first two digits m and n. Then we have mb 2 + nb + (25 - m - n) = 1987. So m(b 2 - 1) + n(b - 1) = 1962 = 18·109. Hence b - 1 divides 18.109. But 109 is prime, so we must have b = 19. We have 1987 = 5·19 2 + 9·19 + 11 and 5 + 9 + 11 = 25, so b = 19 works. 19th CanMO 1987 © John Scholes jscholes@kalva.demon.co.uk 21 June 2002