CAT 2025 Lesson : Factors & Remainders - Operations on Remainders

3.5 Sum of Remainders

Rule: Remainder of Sum of numbers $=$ Sum of remainders when each of the numbers is divided by the divisor.

When $n = a_1 + a_2 + a_3 + ...,$

$\mathrm{Rem} \left(\dfrac{n}{d}\right) = \mathrm{Rem} \left(\dfrac{a_{1} + a_{2} + a_{3} +...}{d}\right) = \mathrm{Rem} \left(\dfrac{a_{1}}{d}\right) + \mathrm{Rem} \left(\dfrac{a_{2}}{d}\right) + \mathrm{Rem} \left(\dfrac{a_{3}}{d}\right) + ...$

If the sum of remainders is greater than $d$, this is once again divided by $d$ to ascertain the final remainder.

3.6 Product of Remainders

Rule: Remainder of the product of numbers $=$ Product of remainders when each of the numbers is divided by the divisor.

When $n = a_1 \times a_2 \times a_3 \times ...$,
$\mathrm{Rem} \left( \dfrac{n}{d} \right) = \mathrm{Rem} \left( \dfrac{a_1 \times a_2 \times a_3 \times...}{d} \right) = \mathrm{Rem} \left( \dfrac{a_1}{d} \right) \times \mathrm{Rem} \left( \dfrac{a_2}{d} \right) \times \mathrm{Rem} \left( \dfrac{a_3}{d} \right) \times ...$

Explanation: Let's say there are three numbers $a_1$, $a_2$ and $a_3$. The remainders obtained when each of $a_1$, $a_2$ and $a_3$ is divided by divisor $d$, are $r_1$, $r_2$ and $r_3$ respectively. And, the quotients are $q_1$, $q_2$ and $q _3$ respectively.

$a_1 = dq_1 + r_1$; $a_2 = dq_2 + r_2$; $a_3 = dq_3 + r_3$

$\mathrm{Rem} \left( \dfrac{a_1 \times a_2 \times a_3}{d} \right) = \mathrm{Rem} \left( \dfrac{(dq_1 + r_1)(dq_2 + r_2)(dq_3 + r_3) }{d} \right)$

In the expansion, all the terms in the numerator is a multiple of $d$, but for the last term, which is $r_1 \times r_2 \times r_3$.

$\therefore$ $\mathrm{Rem} \left(\dfrac{a_1 \times a_2 \times a_3 }{d}\right) = \mathrm{Rem} \left(\dfrac{r_1 \times r_2 \times r_3}{d}\right)$

3.7 Remainder of Exponents

$(a + b)^n = ^{n}{c_0 a^n b^0} + ^{n}{c_1 a^{n - 1} b^1} + ... +$ $^{n}{c_{n-1} a^1 b^{n - 1}} +$ $^{n}{c_n a^0 b^n}$

Let's say a number $a$ when divided by $d$, results in a quotient of $q$ and remainder of $r$.
$a = dq + r$

$\mathrm{Rem} \left(\dfrac{a^n}{d}\right) = \mathrm{Rem} \left(\dfrac{(dq + r)^n}{d}\right)$

$= \mathrm{Rem} \left(\dfrac{^{n}c_0(dq)^n r^0 + ^{n}c_1(dq)^{n - 1}r^1 + .... + ^{n}c_{n - 1}(dq)^1 r^{n - 1} + ^{n}c_n (dq)^0 r^n}{d}\right)$

All terms are divisible by $d$ except for the last term.

$\therefore$ $\mathrm{Rem} \left(\dfrac{(dq + r)^n}{d}\right) = \mathrm{Rem} \left(\dfrac{r^n}{d}\right)$


The following example uses a combination of the rules.