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 !
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 $p=5,q=6$.
Suppose the highest common factor of $A$ and $B$ is $f$
so $A=af$ and $B=bf$ where $a$ and $b$ are coprime
and the lowest common multiple of $A$ and $B$ is $abf$
and thus $A \times B = af \times bf = abf \times f$
For the actual question, it translates into finding $a$ and $b$ such that each is no more than $\dfrac{100}{15}$, they are coprime, and $ab=\dfrac{450}{15}$
