Question

What is the remainder when $4^{60}$ is divided by 7?

• A. 4
• B. 2
• C. 6
• D. 1

Answer 1

The correct option is D 1

To find the cyclicity, we keep finding the remainders till we get a repeating pattern. It can be understood with the following example:
$\begin{array}{ccccccccc}Number& 4^1& 4^2& 4^3& 4^4& 4^5& 4^6& 4^7& 4^8\\ Remainder& 4& 2& 1& 4& 2& 1& 4& 2\end{array}$
Now, $4^4$ gives us the same remainder as $4^1$, so the Cyclicity is of 3(as the remainders start repeating themselves after $4^3).$
So, any power of 3 or a multiple of 3 will give remainder of 1. So, $4^{60}$ will give 1 as the remainder.