Number of Factorials (Limiting Factor)

GRE Quant Problem Solving
1

When we look at the below problem:
How many times would does 24 divide into 40!,
we prime factor 72 and get 2^3 and 3^2, and rightly 2^3 is the limiting factor and the tutor uses the number line as the analogy, where 2^3 occurs every four times and 3^1 would occur every 3 times.

Going with the same logic for another question.
How many times does 90 divide into 45!

We would prime factor 90, and get 10 and 9 and then further division would make 10-> 2^1, 5^1 and 9->3^2,
What is the limiting factor here? It is a 2^1,5^1 and 3^2
2 occurs every 2nd time on the number line, 5 occurs every 5 times on the number line and 3^2 occurs every 6 times on the number line.

But the catch is when we divide 45 by 5, we would get 10, and 3^2 would also occur 10 times or rather go into 45! (45/3 gives 21 3’s, 21/2=10)

What is the limiting factor in this case?

2

If you’re not sure what the “limiting factor” is then you can just compute it for the cases which u can’t make out.

There are 10 5's and 21 3's in 45!, which means you can have 10 groups of 3’s. Since you can have at most 10 copies of either 5 or 9, then the limiting factor can be either one of these.

In particular, you can surmise that x = 10 when calculating the maximum integer value of x in \frac{45!}{90^x} (assuming this was your question) bc of the aforementioned reason (pick either 5 or 9 as your limiting factor and do the usual).