Cross-Posted from [Math Stackexchange] Two positive integers $p,q$ and a prime $r$ are given, such that $r>p>q>1$.<br> I have to show that there exist $n$ such that $$r|\binom{p^n}{q^n}$$ Should I use Lucas' theorem? I can't solve it. [Math Stackexchange]:https://math.stackexchange.com/questions/2475002/show-that-there-exist-n-such-that-r-binompnqn