The greatest integer which divides (p+1)(p+2) | vedclass

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(B) The expression $(p+1)(p+2)(p+3) \ldots (p+q)$ represents the product of $q$ consecutive natural numbers starting from $(p+1)$.We know that the pro

Vedclass · Accepted Answer

$q!$

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(B) The expression $(p+1)(p+2)(p+3) \ldots (p+q)$ represents the product of $q$ consecutive natural numbers starting from $(p+1)$.We know that the product of $q$ consecutive natural numbers is always divisible by $q!$.This is because the expression can be written as $\frac{(p+q)!}{p!} = q! 	imes \binom{p+q}{q}$.Since $\binom{p+q}{q}$ is an integer for all $p, q \in N$,the expression is always divisible by $q!$.Thus,the greatest integer that divides this product for all $p \in N$ is $q!$.

The greatest integer which divides $(p+1)(p+2)(p+3) \ldots (p+q)$ for all $p \in N$ and fixed $q \in N$ is

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