Question

Prove that product of r consecutive positive integers is divisible by r.

Answer 1

The product of r consecutive integer can be

represented as

$(n+r)(n+r-1).....(n+1)=\frac{(n+r)!}{n!}$

where n is the number less than the smallest of

the consecutive integers. Now, if it is true that prime

in $(n+r)!$ appear just as frequently or more as in

$n!r!$ ,then now for same integer k that $(n+r)!=k.n!r!$

so, $\frac{(n+r)!}{n!}=\frac{k.n!r!}{n!}=k.r!$

and is therefore divisible by r!.