# How to Find the Remainder of an Exponential Expression?

Find the remainder of an expression formed by “division of exponential number to a natural number”

## Find the remainder of 7^{700}/100

\mathrm{Find \ the \ remainder \ of \ \ }\dfrac{7^{700}}{100}

## Solution

Let’s check the remainder of the expression when power is lower

\begin{aligned} & \dfrac{7^{0}}{100}=\dfrac{1}{100} \ \Rightarrow \ Remainder = 1\\ \\ & \dfrac{7^{1}}{100}=\dfrac{7}{100} \ \Rightarrow \ Remainder = 7\\ \\ & \dfrac{7^{2}}{100}=\dfrac{49}{100} \ \Rightarrow \ Remainder = 49\\ \\ & \dfrac{7^{3}}{100}=\dfrac{343}{100} \ \Rightarrow \ Remainder = 43\\ \\ & \dfrac{7^{4}}{100}=\dfrac{2401}{100} \ \Rightarrow \ Remainder = 1\\ \\ & \dfrac{7^{5}}{100}=\dfrac{16807}{100} \ \Rightarrow \ Remainder = 7\\ \end{aligned}

Remainders are repeating in a pattern **1, 7, 49, 43, 1, 7,…**

So we can create a specific formula for this expression when power is *n*

\small{Remainder \ of \ \dfrac{7^n}{100}=Remainder \ of \ \dfrac{7^{(Remainder \ of \ \scriptsize{\dfrac{n}{4})}}}{100}}

to solve the question, we need to put *n* = 700

\begin{aligned} & \small{Remainder \ of \ \dfrac{7^{700}}{100}=Remainder \ of \ \dfrac{7^{(Remainder \ of \ \scriptsize{\dfrac{{700}}{4})}}}{100}}\\ \\ & \small{Remainder \ of \ \dfrac{7^{700}}{100}=Remainder \ of \ \dfrac{7^{0}}{100}}\\ \\ & \small{Remainder \ of \ \dfrac{7^{700}}{100}=Remainder \ of \ \dfrac{1}{100}}\\ \\ & \small{Remainder \ of \ \dfrac{7^{700}}{100}=1} \end{aligned}

**Remainder of 7 ^{700}/100 = 1**