Q:

Is this proof of the infinitude of primes valid?

The current issue
(May 2015)
of the American Mathematical Monthly
has a one-line proof
that there are an infinite 
number of primes,
and I don't see why it is correct.
Here is the proof:
If the set of primes
is finite,
then
$$0
< \prod\limits_{p} \sin\left(\frac{\pi}{p}\right)
= \prod\limits_{p} \sin\left(\frac{\pi(1+2\prod_{p'}p')}{p}\right)
=0
.$$
(That's the whole proof.)
I see why the first equality holds,
since, if there are only
a finite number of primes,
$p \mid \prod_{p'}p'$
for all $p$.
But I do not see why
the second equality
("$= 0$")
holds.
None of the terms
in the product
are zero,
and, since there are only
a finite number of them,
the product is not zero.
So,
do I not understand the proof,
or is the proof incorrect?
Thank you.

A:

We must have that $1+2\prod_{p'}p'$ is divisible by some prime $q$, so $1+2\prod_{p'}p' = kq$ for some integer $k$.  But then,
$$\sin\left(\frac{\pi(1+2\prod_{p'}p')}{q}\right) = \sin \pi k = 0$$
which gives the right-hand equality.

