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 $X$?

If one of these divide $X$ (call it $p_{i}$) then we have that the following:
$\frac{p_{1}p_{2}...p_{n}}{p_{i}} + \frac{1}{p_{i}}$
is some integer. Note that the first term is surely an integer (we would just cancel out $p_{i}$) so the second term is also an integer - impossible: none of these numbers divide $1$. 
