I believe to have proven something that I need for an algorithm I'm developing.

The proposition is:

$\displaystyle lcm(b, c) / gcd(a, lm(b, c)) = lcm(a, b, c) / a;$

Proof:

$\displaystyle lcm(b,c) / gcd(a, lcm(b, c)) = lcm(a, lcm(b, c)) / a;$

Substitution: $\displaystyle k:= lcm(b, c);$

$\displaystyle k / gcd(a, k) = lcm(a, k) / a;$

$\displaystyle a \times k = lcm(a, k) \times gcd(a, k);$ // see Greatest common divisor - Wikipedia, the free encyclopedia

Is this correct?

Thank you

Bernhard