Let $a,b,c$ be non-zero integers . Let $(x,y)$ denote the gcd and $[x,y]$ denote the lcm of two non-zero integers . Then is it true that $ (a,[b,c])=[(a,b),(a,c)]$ ?
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I believe so. Take a look at the prime factorization and what gcd and lcm do to the exponents of the primes. – green frog Aug 16 '17 at 21:17
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In addition to the dupe I linked above see also See also this question for the dual. – Bill Dubuque Aug 16 '17 at 21:31