Prove-that-gcd-gcd-A-B-gcd-B-C-gcd-C-A-gcd-A-B-C-

Question Number 33369 by Rasheed.Sindhi last updated on 15/Apr/18

Prove that

\gcd (\gcd (A, B), \gcd (B, C), \gcd (C, A))

= \gcd (A, B, C)

Answered by MJS last updated on 15/Apr/18

set theory

U \cap V = V \cap U

U \cap U = U

U \cap (V \cap W) = (U \cap V) \cap W = U \cap V \cap W

P_{A}, P_{B}, P_{C} : sets of prime factors

of A, B, C

P_{g c d (A, B)} = P_{A} \cap P_{B}

(P_{A} \cap P_{B}) \cap (P_{B} \cap P_{C}) \cap (P_{A} \cap P_{C}) =

= P_{A} \cap P_{B} \cap P_{B} \cap P_{C} \cap P_{A} \cap P_{C} =

= (P_{A} \cap P_{A}) \cap (P_{B} \cap P_{B}) \cap (P_{C} \cap P_{C}) =

= P_{A} \cap P_{B} \cap P_{C} = P_{g c d (A, B, C)}

Commented by Rasheed.Sindhi last updated on 15/Apr/18

e^{x} c e l l e n t S i r! Th α n k s!