Question

What is the remainder when ${{6}^{50}}$ is divided by 216?
A. 1
B. 36
C. 5
D. 214

Answer 1

Hint: To find the the remainder when ${{6}^{50}}$ is divided by 216, we will split ${{6}^{50}}$ using the formulas ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\text{ and }{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ . We will get \[{{\left( {{6}^{3}} \right)}^{16}}\times 6\times 6\] . Now, we can divide this by 216. We can write this as $\dfrac{{{\left( 216 \right)}^{16}}\times 6\times 6}{216}$ . Hence, $\text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}\times 6\times 6}{216} \right]=\text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}}{216} \right]\text{+ Remainder of }\left( \dfrac{6\times 6}{216} \right)...(i)$ . Remainder of $\left[ \dfrac{{{\left( 216 \right)}^{16}}}{216} \right]$ is always 0 . We also know that the remainder of $\dfrac{6\times 6}{216}=\dfrac{36}{216}$ is 36. By substituting these in (i), we will get the required solution.

Complete step by step answer:
We need to find the remainder when ${{6}^{50}}$ is divided by 216.
First let us see how 216 can be reframed.
We can write 216 as
$216={{6}^{3}}$
We know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\text{ and }{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
Now, let us write \[{{6}^{50}}\] as
\[{{\left( {{6}^{3}} \right)}^{16}}\times 6\times 6\]
Now let us divide this by $126$ . We will get
$\dfrac{{{\left( 216 \right)}^{16}}\times 6\times 6}{216}$
Now, let us find the remainder of this.
$\text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}\times 6\times 6}{216} \right]=\text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}}{216} \right]+\text{Remainder of }\left( \dfrac{6\times 6}{216} \right)$
Now, let us consider a term $x$ . When we divide $x$ by $x$ we will get the remainder 0 and quotient 1.
When we divide ${{x}^{2}}$ by $x$, quotient will be $x$ and remainder will be 0.
In general, when we divide ${{x}^{n}}$ by $x$ , where $n=1,2,3,...$ , the remainder will always be 0.
Hence, we can write
$\text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}\times 6\times 6}{216} \right]=0+\text{Remainder of }\left( \dfrac{6\times 6}{216} \right)$
$\Rightarrow \text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}\times 6\times 6}{216} \right]=\text{Remainder of }\left( \dfrac{36}{216} \right)$
We know that when a smaller number is divided by a larger number, the remainder will always be the smaller number itself. That is,
$216\overset{0}{\overline{\left){\begin{align}
  & 36 \\
 & -0 \\
 & \_\_\_ \\
 & 36 \\
\end{align}}\right.}}$
Hence, we can write
$\text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}\times 6\times 6}{216} \right]=36$
Thus, the remainder when ${{6}^{50}}$ is divided by 216 is 36.

So, the correct answer is “Option B”.

Note: We have used the + in the formula $\text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}\times 6\times 6}{216} \right]=\text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}}{216} \right]+\text{Remainder of }\left( 6\times 6 \right)$ . Do not put a $\times $ sign, for example $\text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}\times 6\times 6}{216} \right]=\text{Remainder of }\left[ \dfrac{{{\left( 216 \right)}^{16}}}{216} \right]\times \text{Remainder of }\left( 6\times 6 \right)$ . You must know the rules of exponents as these will be used to split the terms with large exponents.