I read this in a math textbook, stated as obvious fact. I cannot wrap my head around why this works. I understood the division lemma, and also got the algorithm to work: but the proof of why the algorithm works, just said "because of the equation" that I put in the question.
Could you please elaborate on why it works?
1) Suppose that $m$ divides $c$ and $d$. Then $m$ divides $c-dq$, so $m$ divides $d$ and $r$.
2) Suppose that $m$ divides $d$ and $r$. Then since $c=dq+r$, we get that $m$ divides $c$ and $d$.
Thus the set of common divisors of $c$ and $d$ is exactly the same as the set of common divisors of $d$ and $r$.
In particular, the greatest common divisors are the same (and the second greatest, and so on).
Hint $ $ If $\ n\mid d\ $ then $\ n\mid dq+r\!\iff\! n\mid r.\,$ Thus $\,\{d,\,dq+r\}\, $ and $\,\{d,r\}\, $ have the same set $\,S\,$ of common divisors $\,n,\,$ so they have the same greatest common divisor $( = \max\, S).$
Let the HCF of $c$ and $d$ be $k$.
Converting the given equation to modulo k
$$dq+r ≡ 0 \mod k$$
since $c$ is divisible by $k$. But $d$ is also divisble by $k$ and thus,
$$r ≡ 0 \mod k$$
implying $r$ is divisble by $k$.
Therefore $k$ divides both $r$ and $d$ and hence must divide $HCF(r,d)=m$
Similarly it can be shown that $m$ must divide $k$. Thus $k=m$
