Elementary Methods in Number Theory Exercise 1.4.29

Let $a$ and $b$ be positive integers such that
\begin{equation}
a+b=57
\end{equation}
and
\begin{equation}
[a,b]=680
\end{equation}Find $a$ and $b$.

Solve:I try to find $(a,b)$ first,Let $(a,b)=t$,then
\begin{equation}
t|57,t|680
\end{equation}
then $t=1$.So $ab=[a,b](a,b)=680$.So
\begin{align*}
\begin{cases}
a+b=57\\
ab=680\\
\end{cases}
\end{align*}
So
\begin{align*}
\begin{cases}
a=17\\
b=40\\
\end{cases}
\end{align*}or
\begin{align*}
\begin{cases}
a=40\\
b=17\\
\end{cases}
\end{align*}