8th ASU 1974 Problem 2 Find the smallest positive integer which can be represented as 36 m - 5 n .
11 Solution Obviously 11 = 36 1 - 5 2 , and we guess that this is the best possible. We cannot have 36 m - 5 n = k, where 2, 3 or 5 divides k (because then 2 would divide 5 n and similarly in the other cases). So the only possible values of k < 11 are 1, 7. We have 36 m - 5 n = 1 mod 5, so k ≠ 7. Similarly, 36 m - 5 n = 3 mod 4, so k ≠ 1. Thanks to Suat Namli 8th ASU 1974 © John Scholes jscholes@kalva.demon.co.uk 3 March 2004 Last corrected/updated 3 Mar 04