Number Theory: Multiples

First of all, let's find the multiples of 3 in 24!.

1*3, 2*3 ---> 8*3 = 8 multiples

But we can't forget about the multiples of 3². These have an extra 3.

9*1, 9*2 = 2 multiples

8 + 2 = 10

The exponent is 10: 3¹⁰.

it has something to do with number theory because I did it before and I forgot the answer

First, notice that 24! is an enormous number, and it is the product of 24 consecutive integers, so it has a lot of factors.

Also, notice that if we omit numbers that aren't divisible by 3, we have \(3\times 6 \times 9 \times \cdots \times 24\) as a factor of 24!.

We will find the exponent of 3 in this expression because other terms are not divisible by 3.

\(3\times 6 \times 9 \times \cdots \times 24 = (3\times 1) \times (3\times 2) \times (3\times 3) \times \cdots \times (3\times 8)\)

You may notice that the product every 3 consecutive terms is divisible by \(3^4\).

We have 

\((3^4)^2 \times 3^2 | 24!\)

Simplify to get 

\(3^{10} | 24!\)

And we cannot find any other multiple of 3. So the exponent of 3 is 10.