prime numbers and divisibility question - number theory I have a fairly short and straightforward question I would like to ask. Suppose $p_{1},p_{2},...,p_{n}$ are prime numbers. Suppose $X = p_{1}p_{2}...p_{n} + 1$. If $X$ comes out to be a composite number, then why is it true then that none of the $p_{1},p_{2},...,p_{n}$ divide $X$?

Formally assume that $p_i$ divides $X=bp_i+1$ so there exists an integer $a$ such that $X=a.p_i$and then $p_i(a-b)=1$ so $p_i$ divides $1$ which is impossible.
