Why is (2p₁p₂...pₙ)⁴ + 1 divisible by an odd prime?

[buzzmath]

I got stuck on one part of this proof. I'm trying to show that
(2(p1)(p2)...(pn))^4 + 1 is divisible by an odd prime q. Can anyone help with some suggestions?

 

[Muzza]

I take it you left out a bunch of hypotheses, such that the p_i are integer and non-zero. The result then follows immediately from the fundamental theorem of arithmetic.

 

[buzzmath]

p_i are primes and q is an odd prime different from the p_i's

 

[HallsofIvy]

The point of Muzza's statement about the "fundamental theorem of arithmetic" is that every number is divisible by a prime number! All you need to do is show that (2(p1)(p2)...(pn))^4 + 1 is not divisible by any of the pi (what would the remainder be?) and that it is not divisible by 2.