10th APMO 1998 Problem 5 What is the largest integer divisible by all positive integers less than its cube root. Solution
: 420. Let N be a positive integer satisfying the condition and let n be the largest integer not exceeding its cube root. If n = 7, then 3·4·5·7 = 420 must divide N. But N cannot exceed 8 3 - 1 = 511, so the largest such N is 420. If n ≥ 8, then 3·8·5·7 = 840 divides N, so N > 729 = 9 3 . Hence 9 divides N, and hence 3·840 = 2520 divides N. But we show that no N > 2000 can satisfy the condition. Note that 2(x - 1) 3 > x 3 for any x > 4. Hence [x] 3 > x 3 /2 for x > 4. So certainly if N > 2000, we have n 3 > N/2. Now let p k be the highest power of k which does not exceed n. Then p k > n/k. Hence p 2 p 3 p 5 > n 3 /30 > N/60. But since N > 2000, we have 7  N, so N cannot satisfy the condition. 10th APMO 1998 © John Scholes jscholes@kalva.demon.co.uk 11 Apr 2002