I would like to write a Mathematica code to decompose an integer into two or more parts with primes in special intervals.
For instance, I want to decompose $m=\binom{n}{k}$ into two parts U and V such that $\binom{n}{k}=UV$ and $$ U=\prod_{\substack{p\le k\\p^\alpha||m}} p^{\alpha} $$ and $$ V=\prod_{\substack{k<p\le n\\p^\alpha||m}} p^{\alpha} $$ where $p^\alpha||m$ mean $p^\alpha$ is the maximum prime power dividing m.
I tried
Product[If[Element[p,Primes]&&Mod[Binomial[n, k], p] == 0, p, 1], {p,2,k-1}]
Product[If[Element[p,Primes]&&Mod[Binomial[n, k], p] == 0, p, 1], {p,k,n}]
, but it was not true