If the quotient and the rest of the integer division between $b$ and $8$ are $q$ and $k$, respectively, what are the values of the quotient and rest of division between $8$ and $8b-75$?
Work:
If the quotient and the rest of the integer division between $b$ and $8$ are $q$ and $k$ it means $$b=8q+k.\tag{1}\label{1}$$ Let $c,r\in\Bbb{Z}$ be the rest and division between $8$ and $8b-75$ i.e., $$8b-75=8c+r.\tag{2}\label{2}$$
From $\ref{1}$ we observe that $$8q=b-k\implies8\mid(b-k)\tag{3}\label{3}.$$ From $\ref{2}$ we observe that $$8b=8c+r+75\implies8\mid(8c+r+75)\tag{4}\label{4}.$$
$\ref{3}+\ref{4}$ means $$16\mid(8c+r+75+b-k)\tag{5}\label{5},$$ and now I am stuck here.
Also I tried: plugin $\ref{1}$ into $\ref{2}$ we get $$8(8q+k)-75=8c+r\implies64q+8k-75=8c+r\implies\text{???}$$
Am I following the right steps?
By $(1)$ and $(2)$ you have that $$8b-75=8(8q+k)-75=8(8q+k-10)+5.$$ So we get that $$c=8q+k-10, \ r=5$$because $0\leq 5<8.$