9th VMO 1970

Find all positive integers which divide 1890·1930·1970 and are not divisible by 45.

Answer

k·2^a7^b193^c197^d, where k = 1, 3, 3², 3³, 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 = 2·3³5·7, 1930 = 2·5·193, 1970 = 2·5·197 (and 193 and 197 are prime). So 1890·1930·1970 = 2³3³5³7·193·197.

Thanks to Suat Namli

9th VMO 1970

© John Scholes
jscholes@kalva.demon.co.uk
7 March 2004
Last corrected/updated 6 Mar 04