9th VMO 1970 Problem A2 Find all positive integers which divide 1890·1930·1970 and are not divisible by 45.
k·2 a 7 b 193 c 197 d , where k = 1, 3, 3 2 , 3 3 , 5, 3·5, a = 0, 1, 2, or 3, b = 0 or 1, c = 0 or 1, d = 0 or 1 (192 solutions in all) Solution 1890 = 2·3 3 5·7, 1930 = 2·5·193, 1970 = 2·5·197 (and 193 and 197 are prime). So 1890·1930·1970 = 2 3 3 3 5 3 7·193·197. Thanks to Suat Namli 9th VMO 1970 © John Scholes jscholes@kalva.demon.co.uk 7 March 2004 Last corrected/updated 6 Mar 04