Let m be the least common multiple of a,b and c be another common multiple. Prove that m divides c.

Question

I have tried to solve the problem. I have started by assigning m=(ab)/d, where d is the greatest common divisor of a and b. Then I set up c = aq. That is where I am stuck. I am really not sure what to do next.

Answer 1

Hint: $c = rm+d$ where $r$ is some integer and $d < m$. So $d = c-rm$. As $a$ divides $c$ and $m$, and $-r$ is an integer, $a$ divides $d$. Likewise as $b$ divides both $c$ and $m$, it follows that $b$ divides $d$. So both $a$ and $b$ divide $d$. But $m$ is LCM of $a$ and $b$ and $d< m$. What can you say about $d$?