Problem 5
Determine the largest integer n with the property that n is divisible by every positive integer less than the cube root of n.
Step 3 of 4: Use four consecutive divisors
Detailed analysis
If N is not a cube, then t,t-1,t-2,t-3 are all less than the cube root of N and hence divide N. If N=t^3, then t divides N directly, while t-1,t-2,t-3 are less than the cube root and divide N. Thus in either case the product is divisible by 8, and the possible common factors among four consecutive integers contribute at most a factor 6; consequently their product divides 6N. In particular it is at most 6N.