1st Mexico 1987 Problem A2 How many positive integers divide 20! ?
2 4 3 3 5·19 = 41040 Solution 20! is divisible by [20/2] + [20/4] + [20/8] + [20/16] = 18 powers of 2, and [20/3] + [20/9] = 8 powers of 3. It is divisible by 54, 72, 11, 13, 17 and 19. In other words, 20! = 2 18 3 8 5 4 7 2 11·13·17·19. So it has (18+1)(8+1)(4+1)(2+1)(1+1)(1+1)(1+1)(1+1) = 2 4 3 3 5·19. 1st Mexican 1987 © John Scholes jscholes@kalva.demon.co.uk 21 February 2004 Last corrected/updated 21 Feb 04