An algebra problem by Kartik Jay

Let $n$ be the dividend

By Euclid's Division Algorithm,

$\large{n=1995q +75\quad{\text{where q is the quotient}}}$

Plugging in values for $q$ we get that ,

$\large{n=2070 \quad\text{when n = 1}}$

Applying this value in a division algorithm with $57$ we get;

$2070= 57\times 36 + 18$

From this we get the Remainder , $\Large{18}$

Plugging any value of $q$ and subsequent value of $n$ we find that the remainder always remain the same.

$\large{\text{This is because 57 here is a factor of 1995}\quad\text{57 x 35 = 1995}}$

Thus we can generalize ,that if

$\large{n =b_1q_1 + r_1}$

$\large{n =b_2q_2 + r_2}$

$\large{If\quad b_2\times x =b_1 \quad where;\quad x\in \Bbb{Z^{+}}}$

$\large{then\quad r_2 = r_1 - b_2 \quad where;\quad b_2\lt{b_1}}$