Suppose a,b, p Z and p is prime. Prove that if p | ab then p | a or p | b. Show it suffices to prove: Ifpabandpa, thenpb. In what follows, supposepis...

Suppose a,b, p ∈ Z and p is prime. Prove that if p | ab then p | a or p | b.

• 1. Show it suffices to prove: If p∣ab and p∤a, then p∣b.
• In what follows, suppose p is prime, p∣ab and p∤a.
• 2. Show gcd(a,p)=1.
• 3. From (2) and the proposition on page 152, there exists k,ℓ∈Z such that ak+pℓ=1.
• 4. Invoke the assumption p∣ab and use a bit of algebra to show p∣b, completing the proof.