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 2 of 4: Force t to be at least 13
Detailed analysis
Assume N>420 and put t=floor(cube root of N). Then t is at least 7. Since 420 divides N, N is at least 840, so t is at least 9. Now 2520=lcm(1,...,9) divides N, hence N is at least 2520 and therefore t is at least 13. Also N<(t+1)^3 by the definition of t.