LCM Inequality proof

Question

i have a question that i cannot solve

let a,b,c be positive integers such that b divides c prove that

LCM(a,b) <= LCM(a,c)

so far i tried to go with the definition of gcd and division but it does not help because i cannot prove that gcd(a,b) > gcd(a,c) so if anyone can give me a lead or a tip that would be great!

Thank you for your help.

Answer 1

$\mathrm{lcm}(a,c)$ is by definition a multiple of $a$ and $c$. Because $b|c$, it is also a multiple of $a$ and $b$. By definition of $\mathrm{lcm}(a,b)$ as the least common multiple of $a$ and $b$, you have directly $$\mathrm{lcm}(a,b) \leq \mathrm{lcm}(a,c)$$