# Find the largest integer $k$ for which $3^k$ divides $400 \choose 200$.

I was working through a problem set for number theory and needed some help with this problem:

Find the largest integer $$k$$ for which $$3^k$$ divides $$400\choose 200$$.

I know this will reduce to $$\frac{400!}{(200!)^2}$$ and I found that the largest power of $$3$$ that divides $$400!$$ is $$196$$ and the largest power that divides $$200!$$ is $$97$$ but I don't know how to put them together.

• $$196-2\cdot97=?$$ – lab bhattacharjee May 7 at 15:02

You have found that $$400!=3^{196}m$$ and $$200!=3^{97}p$$ where $$m,p$$ are integers not divisible by $$3$$. Then $$\frac {400!}{200!^2}=\frac {3^{196}m}{(3^{97}p)^2}=\frac {3^{196}m}{3^{2\cdot97}p^2}=3^2\frac m{p^2}$$