Let a∣c and b∣c such that greatest common divisor (gcd) gcd(a,b)=1, Show that ab∣c.
Answer
HINT a,b | c ⇒ ab | ac,bc ⇒ ab | (ac,bc)=(a,b)c=c via (a,b)=1.
NOTE This proof works in every domain where GCDs exist, since it doesn't use Bezout's identity. Instead it uses the GCD distributive law (ac,bc)=(a,b)c, true in every GCD domain, viz.
LEMMA (a,b) = (ac,bc)/c if (ac,bc) exists.
Proof d | a,b ⟺ dc | ac,bc ⟺ dc | (ac,bc) ⟺ d | (ac,bc)/c
The above proofs use the universal definitions of GCD, LCM, which often serve to simplify proofs, e.g. see this proof of the GCD * LCM law.
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