Question

The remainder when ${23^{23}}$ is divided by $53$ is
${\text{(A) 17}}$
${\text{(B) 21}}$
${\text{(C) 30}}$
${\text{(D) 43}}$

Answer 1

Hint: Here we have to find the remainder of the given number. Using the property of modulo, we get the value for finding the remainder. Then doing some simplification we get the required answer.

Formula used:
Properties of modulo: ${({a^b})^c}$ can be written as ${a^{bc}}$

Complete step-by-step answer:
We have to find the remainder when ${23^{23}}$ is divided by $53$
To find the remainder we have to divide the \[2\] numbers, this can be written as:
\[ \Rightarrow \dfrac{{{{23}^{23}}}}{{53}}\]
Now in the above terms the numerator can be expanded and written as:
\[ \Rightarrow \dfrac{{{{23}^{22}} \times 23}}{{53}}\]
Now since we know the property that ${({a^b})^c}$ can be written as ${a^{bc}}$ we split the numerator and write is as:
\[ \Rightarrow \dfrac{{{{({{23}^2})}^{11}} \times 23}}{{53}}\]
Now on squaring the bracket term and we get:
\[ \Rightarrow \dfrac{{{{(529)}^{11}} \times 23}}{{53}}\]
Now since the question is to find the remainder of the term, we will divide $529$ with the denominator value which is $53$.
On dividing we get:
Since the remainder when $529$ is divided by $53$ is $52$, since $52$ can be written as $53 - 1$ we can write that the remainder is $ - 1$.
On re-writing the equation we get:
\[ \Rightarrow \dfrac{{{{( - 1)}^{11}} \times 23}}{{53}}\]
Now since $ - 1$ is raised to a power which is odd, the equation can be simplified and re-written as:
\[ \Rightarrow \dfrac{{ - 23}}{{53}}\]
Now to find the remainder we have to find the remainder therefore it can be found by:
\[ \Rightarrow 53 - 23\]
On subtracting we get:
\[ \Rightarrow 30\]
Therefore, the remainder is $30$,

Therefore, the correct option is $(C)$.

Note:
When there are such high powers and calculations have to be done then we use modulo to get the answers because a very powerful calculator is required to compute such high powers.
The mod function is generally expressed as: $r = a\bmod b$,
Here $r$ is the remainder or the modulus value, $a$ is the dividend and $b$ is the divisor.
For example $10\bmod 3 = 1$ since the remainder we get after dividing $10$ by $3$ is$1$.