elementary number theory - Proving property of congruence - help needed

Let $c,d,m,k ∈ \mathbb{Z}$ such that $m ≥ 2$ and $k$ is not zero. Let
$f = \gcd(k,m)$. If $c \equiv d \pmod m $ and $k$ divides
both $c$ and $d$, then $$ \frac{c}{k} \equiv \frac{d}{k}
\left({\bmod} \frac{m}{f}\right)$$

My lecturer asked me to prove this statement, as an exercise.

To prove this, I started by considering the two cases:

1. If suppose $k$ and $m$ are relatively prime, meaning that the $f=1$, then by the Congruence and Division Cancellation Law, we
   know that $ \frac{c}{k} \equiv \frac{d}{k} \pmod m$. For
   this case, $ \frac{c}{k} \equiv \frac{d}{k} \big({\bmod}
   \frac{m}{f}\big)$ must be true since $\frac{m}{f} = \frac {m}{1} =
   m$


2. Now, it remains to prove the other case, where $k$ and $m$ are not relatively prime. By the definition of divisibility, we know that
   $c \equiv d \pmod m $ is equivalent of saying $c =
   d+mj$. We divide both sides by the common divisor k, gives us
   $\frac{c}{k} = \frac{d}{k} + \frac{mj}{k}$. Now, we consider $f = \gcd(k,m)$. This implies that there must exists and integer i, such that $k=lf$, for some integer $l$. Thus, $\frac{c}{k} = \frac{d}{k} + \frac{mj}{k} = \frac{d}{k} + \frac{mj}{lf}$. (Is this true? -- since $l$ does not divide $m$ and the fact that it has to be integer, then $l$ must divide $j$)

I have no idea where to continue. Am I on the right track? Any hints to finish the proof?

Answer

Since $(k,m)=f$, we get $(k/f,m)=1$ so there is $x$ such that $(k/f)\cdot x\equiv 1\pmod m$ holds. Especially, we get $k\cdot(x/f)\equiv 1\pmod {m/f}$.

From $cx\equiv dx\pmod m$ we can derive $c\cdot (x/f)\equiv d\cdot (x/f)\pmod {m/f}$ and you can check that $c\cdot(x/f)\equiv c/k\pmod {m/f}$.