Proving the Identity: gcd(a, lcm(b,c))

[Dragonfall]

Is this identity true?

gcd(a, lcm(b,c)) = lcm(gcd(a,b), gcd(a,c))

 

[nocturnal]

What have you tried so far?

Have you looked for counter examples or attempted a proof?

Do you have an idea of how to prove/disprove statements like these?

 

[Dragonfall]

I have already proven it, since the exponents of the all primes of the canonical factorization of a, b and c obeys min(a,max(b,c))=max(min(a,b),min(a,c)).