What is the highest power of 12 in 54

The highest power of 12 in 54 factorial (54!) is 26—meaning 12^26 divides evenly into 54!, but 12^27 does not. To find this, you need to count the minimum of the powers of 12's prime factors (12 = 2² × 3) that appear in 54!: specifically, you need pairs of twos and one three for each factor of 12.

The calculation requires counting powers of 2 and 3 in 54! separately using Legendre's formula. For twos: ⌊54/2⌋ + ⌊54/4⌋ + ⌊54/8⌋ + ⌊54/16⌋ + ⌊54/32⌋ = 27 + 13 + 6 + 3 + 1 = 50. For threes: ⌊54/3⌋ + ⌊54/9⌋ + ⌊54/27⌋ = 18 + 6 + 2 = 26. Since 12 = 2² × 3, each factor of 12 requires two factors of 2 and one factor of 3. We have 50 twos (providing 25 pairs) and 26 threes, so the limiting factor is threes at 26, meaning the highest power of 12 in 54! is 26. This type of calculation appears in number theory when analyzing factorial divisibility, in combinatorics when simplifying binomial coefficient calculations, and in algorithm analysis when determining how many times a number can be factored from a product. Understanding prime factorization power-counting is fundamental for working with divisibility in factorial-based formulas.