+0 What’s the smallest n > 1 such that n! is divisible by n^4 ? Explain please
+129895 n = 12
To see why
12! =
12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2
Regrouping, we have
12 * (8 * 6) * (4 * 3) * 9 * 11 * 10 * 7 * 5 * 2 =
12 * (12 *4) * (4 * 3) * (9) * 11 * 10 * 7 * 5 * 2 =
12 * (12) * (4 *3) * ( 9 *4) * 11 * 10 * 7 * 5 *2 =
12 * 12 * 12 * (12 * 3) * 11 * 10 * 7 * 5 * 2 =
12 * 12 * 12 * 12 * 3 * 11 * 10 * 7 * 5 * 2
12^4 * 23100
So
12! /12^4 =
12^4 * 23100 / 12^4 = 23100
