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Question

# What is the highest power of 12 that divides 54!?

CAT 2021

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##### 4
Solution

Correct option is

(A)

12 = 2^2 * 3, so we need to count the highest power of 2 and highest power of 3 that will divide 54! and then we can use this to find the highest power of 12.

54! is a multiple of 2^50 * 3^26. Importantly, these are the highest powers of 2 and 3 that divide 54!.

2^2* 3 = 12. We need to see what is the highest power of 22 * 3 that we can accommodate within 54!

In other words, what is the highest n such that (2^2*3)^n can be accommodated within 2^50* 3^26

Let us try some numbers, say, 10, 20, 30

(2^2*3)^10 = 2^20 * 3^10, this is within 2^50* 3^26

(2^2*3)^20 = 2^40 * 3^20, this is within 2^50* 3^26

(2^2*3)^30 = 2^60 * 3^30, this is not within 2^50* 3^26

The highest number possible for n is 25.

(2^2*3)^25 = 2^50 * 3^25, this is within 2^50* 3^26,

but (2^2 *3)^26 = 2^52 * 3^26, this is not within 2^50 * 3^26

So, 54! can be said to be a multiple of (2^2 * 3)^25.

Or, the highest power of 12 that can divide 54! is 25. Note: For most numbers, we should be able to find the limiting prime. As in, to find the highest power of 10, we need to count 5s.