Problems based on Numbers Test 2

For natural number,
$\frac{\mathit{p}}{\mathit{d}}=\mathit{q}+\mathit{r}$
$\mathit{p}=\mathit{d}\mathit{q}+\mathit{r}$...(i)
$\frac{\mathit{q}}{\mathit{d}\prime }=\mathit{q}\prime +\mathit{r}\prime$
$\mathit{q}=\mathit{d}\prime \mathit{q}\prime +\mathit{r}\prime$......(ii)
By substituting the value of q in equation (i) we get,
$\mathit{p}\mathit{d}\left(\mathit{d}\prime \mathit{q}\prime +\mathit{r}\prime \right)+\mathit{r}$
$\mathit{p}=\mathit{d}\mathit{d}\prime \mathit{q}\prime +\mathit{d}\mathit{r}\prime +\mathit{r}$ ......(iii)
By using equation (iii), we can say that when P is divided by d, then quotient is q’ and the remainder is r + r’d.