i have a question about the following assertion:

let $n,j,u $ positif integer satisfying

$ n \geq 5,$ $ 1\leq j \leq n-1$,$ \; n+1 \leq u \leq n+j$

let $ d[n]:=lcm[1,2,..,n]$ thus $u$ divide $d[n].C_{n+j}^n$

i think i have find a proof using valuation p-addic of prime number appairing in $u$ but i would like have another proof... thanks for help