Find two numbers, $A$ and $B$, both smaller than $100$, that have a lowest common multiple of $450$ and a highest common factor of $15$.
I know that this involves the formula of
$A × B = LCM × HCF$
But I don't quite understand the above formula so I rather memorise it and that is why I can't apply it now. Can anyone explain on how this formula is derived ? Thanks alot in advance !
Best Answer
Your numbers are $15p,15q$ with $\gcd(p,q)=1$ and $1\le p<q<7$. Now $p$ can't be $1$ as this would imply $\text{lcm}(15p,15q)=15q<105$. If $p=2$, $q$ has $2$ choices $3,5$ both result in lcm<$450$. If $p=3$, $q$ has $2$ choices $4,5$ and both result in lcm<$450$. If $p=4$, $q=5\implies$ lcm$=300$. Therefore $p=5,q=6$.