The largest n ∈ N such that 3ⁿ divides 50 ! is :

Solution

Maths sir · 3 steps

Do not list 50 numbers; count how many are divisible by 3, then by 9, then by 27.

1. Count the threes. The power of 3 in 50! is the total number of factors of 3 across 1, 2, …, 50. A multiple of 3 gives at least one, a multiple of 9 gives one more, and a multiple of 27 gives one more again.

   ⌊ 50/3 ⌋ + ⌊ 50/9 ⌋ + ⌊ 50/27 ⌋ + ⌊ 50/81 ⌋

2. Each term. There are 16 multiples of 3, 5 multiples of 9 and 1 multiple of 27 (that is 27 itself). Since 81 > 50, every later term is 0.
3. So. 16 + 5 + 1 = 22, so 3²² divides 50! but 3²³ does not. The answer is (a) 22.

The idea to keep. The power of a prime p in n! is ⌊ n/p ⌋ + ⌊ n/p² ⌋ + ⌊ n/p³ ⌋ + … Keep adding terms until pᵏ is bigger than n.

Answer from the NTA final answer key. Solution by PrepWiser.