I'm going to write $(m,n)$ for $\text{gcd}(m,n)$ throughout what follows.
You want to show that $(a,c)(b,c) \mid ab, c$. From the definition of $(m,n)$, it's easy to show that $(a,c)(b,c) \mid ab$ (since $(a,c) \mid a$ and $(b,c) \mid b$). So it really boils down to showing that $(a,c)(b,c) \mid c$.
Since $(a,c), (b,c) \mid c$, you can write $c = r(a,c) = s(b,c)$ for some integers $r$ and $s$. Therefore, to show that $(a,c)(b,c) \mid c$, it's enough to show that $(b,c) \mid r = c/(a,c)$. This is equivalent to showing that $p \nmid (a,c)$ for any prime number dividing $(b,c)$. This follows from $a$ and $b$ being coprime. (Why?)
Hint $\,\ a,b\mid c\,\Rightarrow\, ab\mid ca,cb\,\Rightarrow\, ab\mid (ca,cb) = c(a,b)\, $ by the GCD Distributive Law (below).
Below are four proofs of the GCD Distributive Law $\rm\:(ax,bx) = (a,b)x\:$ using various approaches: Bezout's identity, the universal gcd property, unique factorization, and induction.
First we show that the gcd distributive law follows immediately from the fact that, by Bezout, the gcd may be specified by linear equations. Distributivity follows because such linear equations are preserved by scalings. Namely, for naturals $\rm\:a,b,c,x \ne 0$
$\rm\qquad\qquad \phantom{ \iff }\ \ \ \:\! c = (a,b) $
$\rm\qquad\qquad \iff\ \: c\:\ |\ \:a,\:b\ \ \ \ \ \ \&\ \ \ \ c\ =\ na\: +\: kb,\ \ \ $ some $\rm\:n,k\in \mathbb Z$
$\rm\qquad\qquad \iff\ cx\ |\ ax,bx\ \ \ \&\ \ \ cx = nax + kbx,\ \,$ some $\rm\:n,k\in \mathbb Z$
$\rm\qquad\qquad { \iff }\ \ cx = (ax,bx) $
The reader familiar with ideals will note that these equivalences are captured more concisely in the distributive law for ideal multiplication $\rm\:(a,b)(x) = (ax,bx),\:$ when interpreted in a PID or Bezout domain, where the ideal $\rm\:(a,b) = (c)\iff c = gcd(a,b)$
Alternatively, more generally, in any integral domain $\rm\:D\:$ we have
Theorem $\rm\ \ (a,b)\ =\ (ax,bx)/x\ \ $ if $\rm\ (ax,bx)\ $ exists in $\rm\:D.$
Proof $\rm\quad\: c\ |\ a,b \iff cx\ |\ ax,bx \iff cx\ |\ (ax,bx) \iff c\ |\ (ax,bx)/x\ \ \ $ QED
The above proof uses the universal definitions of GCD, LCM, which often served to simplify proofs, e.g. see this proof of the GCD * LCM law.
Alternatively, comparing powers of primes in unique factorizations, it reduces to the following
$$\begin{eqnarray} \min(a+x,\,b+x) &\,=\,& \min(a,b) + x\\
\rm expt\ analog\ of\ \ \ \gcd(a \,* x,\,b \,* x)&=&\rm \gcd(a,b)\,*x\end{eqnarray}\qquad\qquad\ \ $$
The proof is precisely the same as the prior proof, replacing gcd by min, and divides by $\,\le,\,$ and using the universal property of min instead of that of gcd, i.e.
$$\begin{eqnarray} {\rm employing}\quad\ c\le a,b&\iff& c\le \min(a,b)\\
\rm the\ analog\ of\quad\ c\ \, |\, \ a,b&\iff&\rm c\ \,|\,\ \gcd(a,b) \end{eqnarray}$$
Then the above proof translates as below, $\ $ with $\,\ m(x,y) := {\rm min}(x,y)$
$c \le a,b \!\iff\!c\!+\!x \le a\!+\!x,b\!+\!x\!$ $\!\iff\! c\!+\!x \le m(a\!+\!x,b\!+\!x)\!$ $\!\iff\!\! c \le m(a\!+\!x,b\!+\!x)\!-\!x$
Theorem $\ \ $ If $\ a,b,x\ $ are positive naturals then $\ (ax,bx) = (a,b)x $
Proof $\ $ By induction on $\color{#0a0}{{\rm size}:= a\!+b}.\,$ If $\,a=b,\,$ $\,(ax,bx) = (ax,ax) = (a)x = (a,b)x\,$ hence it is true. $ $ Else $\,a\neq b;\,$ wlog, by symmetry, $\,a > b\,$ so $\,(ax,bx) = (ax\!-\!bx,bx) = \color{}{((a\!-\!b)x,bx)}\,$ with smaller $\rm\color{#0a0}{size}$ $\,(a\!-\!b) + b = a < \color{#0a0}{a\!+b},\,$ therefore $\,((a\!-\!b)x,bx)\!\underset{\rm induct}=\! (a\!-\!b,b)x = (a,b)x$.
For completeness below we present a proof of the LCM Distributive Law with notation $\,[x,y] := {\rm lcm}(x,y),\,$ using the LCM universal property and basic divisibility properties.
Theorem $\ \ [ax,bx] = [a,b]x $
Proof $\ \ [ax,bx]\mid c\iff ax,bx\mid c\iff a,b\mid c/x\iff [a,b]\mid c/x\iff [a,b]x\mid c$
Therefore $\ [ax,bx] = [a,b]x\,$ since they divide each other by above.
Best Answer
From your work, you can conclude that
$$l = (kl) s + m (lt)$$
Now $m | kl$ by assumption, so $m$ divides the right hand side of this equation. Hence it divides the left.