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