What is the largest integer n such that 12ⁿ divides 50! (50 factorial)?

12ⁿ = 2²ⁿ × 3ⁿ, so we need enough 2s and enough 3s in 50!.

Powers of 2 in 50!: 25 + 12 + 6 + 3 + 1 = 47 ⇒ at most ⌊47/2⌋ = 23 twelves.
Powers of 3 in 50!: 16 + 5 + 1 = 22 ⇒ at most 22 twelves.

The scarcer resource decides: n = 22.

Trap: 23 comes from checking only the 2s; 16 counts only multiples of 3 once.