Question

The greatest integer which divides $(p+1)(p+2)(p+3)....(p+q)$ for all $pϵN$ and fixed $qϵN$ is

• A. $p!$
• B. $q!$
• C. $p$
• D. $q$

Answer 1

Answer: (B)

$q!$

$(p+1),(p+2),(p+3),....(p+q)$ are $q$ consecutive numbers.

$(p+1)(p+2)(p+3)....(p+q)$ is product of ${}^{\prime }{q}^{\prime }$ consecutive natural numbers.

The product will be divisible by all its factors and since it is a product of consecutive natural numbers, so it will always be divisible by $q!$.

Also, $q!$ is the greatest integer amongst all their divisors.

Hence, the greatest integer which divides the product is $q!$.