All Remainder Theorems: Concepts and Solved Answers

Concepts of various remainder theorems is one of the important part of CAT, MAT, GMAT, SSC etc. This is a very vital part for college campus placement apti round.

Jump ahead to

Remainder Theorem

Question: Let N = 1421 × 1423 × 1425
What is the remainder when N is divided by 12? [CAT-2000]
(a) 0
(b) 9
(c) 3
(d) 6

Ans: (c) 3

Explanation:

1421 mod 12 = 5
$\begin{align*} \frac{\text{N}}{12} &= \frac{1421\times 1423\times 1425}{12}\\ &=\frac{5\times 7\times 9}{12}\\ &=\frac{-1\times 9}{12}\\ &=\frac{-9}{12}\\ &=\frac{3}{12}\\ \end{align*}$

Question: $\displaystyle \frac{2851\times (2862)^2\times (2873)^3}{23}$ , Remainder?
(a) 8
(b) 10
(c) 17
(d) 18

Ans: (d) 18

Explanation:

2851 mod 23 = -1

\begin{align*} \frac{2851\times (2862)^2\times (2873)^3}{23} &= \frac{-1\times (10)^2\times (-2)^3}{23}\\ &=\frac{100\times 8}{23}\\ &=\frac{8\times 8}{23}\\ &=\frac{64}{23}\\ &=\frac{18}{23}\\ \end{align*}

Polynomial Remainder Theorem

Question: $\displaystyle \frac{79^6}{11}$ , Remainder?
(a) 0
(b) 1
(c) 8
(d) 9

Ans: (d) 9

Explanation:

\begin{align*} \frac{79^6}{11} &= \frac{(77+2)^6}{11}\\ &=\frac{2^6}{11}\\ &=\frac{64}{11}\\ &=\frac{9}{11} \end{align*}

Question: $\displaystyle \frac{53^{12}}{17}$ , Remainder?
(a) 0
(b) 1
(c) 15
(d) 16

Ans: (d) 16

Explanation:

\begin{align*} \frac{53^{12}}{17} &= \frac{(51+2)^{12}}{17}\\ & =\frac{2^{12}}{17}\\ & = \frac{(2^4)^3}{17}\\ &= \frac{(16)^3}{17}\\ & = \frac{(17-1)^3}{17} \\ &= \frac{(-1)^3}{17}\\ & = \frac{-1}{17}\\ &= \frac{-1+17}{17}\\ &= \frac{16}{17}\\ \end{align*}

Question. $\displaystyle\frac{2^{196}}{96}$ , Remainder?
(a) 32
(b) 64
(c) 16
(d) 2

Ans: (b) 64

Explanation:

$96=32\times 3=2^5 \times 3$
$\begin{align*} \frac{2^{196}}{96} &= \frac{2^5 \times 2^{191}}{2^5 \times 3}\\ &=\frac{(3-1)^{191}}{3}\\ &=\frac{-1}{3}\\ &=\frac{2}{3}\\ &=\frac{64}{96}\\ \end{align*}$

Question. $\displaystyle\frac{629^{24}}{21}$ , Remainder?
(a) 0
(b) 1
(c) 2
(d) 3

Ans: (b) 1

Explanation:

629 \text{ mod } 21 = -1

\displaystyle\frac{629^{24}}{21}=\displaystyle\frac{(-1)^{24}}{21}=\displaystyle\frac{1}{21}

So, remainder =1

Question: The remainder when $7^{84}$ is divided by $342$ [CAT-1999]
(a) 0
(b) 1
(c) 49
(d) 341

Ans: (b) 1

Explanations:

$7^3=343=342+1$
$\begin{align*} \frac{7^{84}}{342} &= \frac{(7^3)^{28}}{342} \\ &= \frac{(342+1)^{28}}{342}\\ &=\frac{1^{28}}{342}\\ &=\frac{1}{342} \end{align*}$

Hence, remainder = 1.

Question: $\displaystyle\frac{7^{84}}{2400}$ , Remainder?
(a) 1
(b) 2399
(c) 4
(d) 7

Ans: (a) 1

Explanations:

Let’ find out which power of 7 is nearer to 2400

\begin{align*} 7^1&=7\\ 7^2&=49\\ 7^3&=343\\ 7^4&=2401\\ \end{align*}

\begin{align*} \frac{7^{84}}{2400} &= \frac{(7^4)^{21}}{2400} \\ &= \frac{(2400+1)^{21}}{2400}\\ &=\frac{1^{21}}{2400}\\ &=\frac{1}{2400} \end{align*}

Hence, Remainder = 1.

Fermat’s Theorem

Question: Which one of the following is the remainder when $10^{20}$ is divided by 7? [UPSC CAPF – 2018]
(a) 1
(b) 2
(c) 4
(d) 6

Ans: (b) 2

Explanations:

\displaystyle\frac{10^{20}}{7} = \displaystyle\frac{(7+3)^{20}}{7}=\displaystyle\frac{3^{20}}{7}

Now, (3, 7) ⟶ Co-prime

So we could apply Fermat’s Theorem

Euler no of 7, ε(7) = 7 – 1 = 6

\displaystyle\frac{3^{20}}{7} = \displaystyle\frac{3^{6\times3 +2}}{7} = \displaystyle\frac{3^{2}}{7}= \displaystyle\frac{9}{7}= \displaystyle\frac{2}{7}

Hence, remainder = 2

Advanced Problems

Question: Find the remainder when $10^{{20}^{30}}$ is divided by 23?
(a) 5
(b) 8
(c) 12
(d) 13

Ans: (d) 13

Explanation:

$\displaystyle \frac{10^{{20}^{30}}}{23}$ , Remainder?

This is going to be a lengthy solution as we would do every step.

$\displaystyle \frac{10^{{20}^{30}}}{23}$ , here (10, 23) are co-prime, so we could apply Fermat’s Theorem

Euler number of 23 is 22.

So let’s get remainder first for this expression, $\displaystyle\frac{20^{30}}{22}$

\begin{align*} \frac{20^{30}}{22}&=\frac{(22-2)^{30}}{22}\\ &=\frac{2^{30}}{22}\\ &=\frac{2^{29}}{11}\\ &=\frac{2^9}{11}\\ &=\frac{512}{11}\\ &=\frac{6}{11}\\ &=\frac{12}{22} \end{align*}

Now our actual problem reduces to $\displaystyle\frac{10^{12}}{23}$ , Remainder?

As the denominator involves 23, let’s find out some initial multiples of it, so that it’s easy to know which power we should work with

23 ⟶ 23, 46, 69, 92, 115, 138, etc.

\begin{align*} \frac{10^{12}}{23}&=\frac{(10^2)^6}{23}\\ &=\frac{(92+8)^6}{23}\\ &=\frac{8^6}{23}\\ &=\frac{64^3}{23}\\ &=\frac{(69-5)^3}{23}\\ &=\frac{-5^3}{23}\\ &=\frac{-125+138}{23}\\ &=\frac{13}{23} \end{align*}

Join Us

Join the discussion and ask any question of maths and reasoning directly to us and other like-minded user just like you!

👉 More on Mathematics for Competitive Exam

<<Prev

Home

Next>>

Please share this post with someone who might find it helpful.

Remainder Theorem

Polynomial Remainder Theorem

Fermat’s Theorem

Advanced Problems

Join Us

RELATED POSTS