halp!

What is the largest integer N such that 7^N = (1000!)/(500!)^2?

I think there is something missing.
You only have one unknown and an = sign therefore there is just one answer.
maybe it should be > or < ?

Also, (1000!)/(500!)^2 is a huge number. I've tried to do it on a couple of calculators and they just give error messages or do nothing at all. I've never seen that before. I am assuming that they just won't handle factorials of this size. The question can still be answered but not simplified.
This might not matter if the question is a bit different.

Sorry! I meant "What is the largest integer N such that 7^N divides (1000!)/(500!)^2.

This question is making my mind boggle.

I suggest that you try doing something very similar only with much smaller numbers and see if you can make sense of it that way.

Maybe you have already done this.

I haven't given up on it but I have other things to do now.

I just couldn't leave it alone.

What is the largest integer N such that 7^N = (1000!)/(500!)^2?

I experimented with 20!/(10!10!) with 3^N
and 30!/(15!15!) with 3^N
If you experiement with thes smaller ones hopefully you will be able to see where I got my logic from.

and this is what I came up with
numerator
1000/(7^1) = 142 (i don't care about the left overs)
1000/(7^2) = 20
1000/(7^3) = 2
1000/(7^4) = 0
Add them all up and you get 164 So 7^164 is a factor of the top

denominator
500/(7^1)= 71
500/(7^2)= 10
500/(7^3)= 1
500/(7^4)= 0
add them up and multiply by 2 = 2*82 = 164 so 7^164 is a factor of the bottom

When you cancel all the 7s cancel out and you get 1
So the biggest power of 7 is 0. (It is not divisable by 7)

I am pretty sure that is correct.