Prove that if p 2 is an integer with the property that for e

Prove that if p 2 is an integer with the property that for every pair b,c of integers p | bc implies that p | b or p | c, then p is prime.

Solution

lets assume p is not prime

=>

there exists integers a,b>1 such that p = ab,

=>
p|ab

=>

p |a or p|b (from hypothesis)

which is impossible since a,b <p

=>

p is a prime

thus proved