WHAT IS THE CORRECT ANSWER

Fixed.

The OA is 12: A.

Quick way to calculate:
51^1 / 13 --> remainder is 12
51^2 / 13 --> remainder is 1
51^3 / 13 --> remainder is 12

There is a cycle of 2
The power 25 is not divisible by 2 and has a remainder of 1.
Therefore, the remainder of 51^25 / 13 is equal to the remainder of 51^1 / 13 --> 12

Answer A

We need to find When \(51^{25}\) is divided by 13, the remainder obtained is

Now, to solve this problem we will be using the concept of Binomial Theorem

We will try to split 51 into two numbers. One number will be a multiple of 13 and will be very close to 51 and other number will be a very small number.

\(51^{25}\) = \((52 - 1)^{25}\) (52 = 13*4)

Now, when we open this using Binomial Theorem then all terms apart from the last term will be a multiple of 52 => a multiple of 13 => Remainder by 13 will be 0

=> Question is reduced to what is the remainder when the last term ( 25C25 \((-1)^{25} * 52^0\) is divided by 13

=> Remainder when -1 is divided by 13
=> -1 + 13 = 12

So, Answer will be A
Hope it helps!

Watch the following video to Learn about Binomial Theorem and How to Solve Similar problems

this way so fast when apply with a^n divide by a number!! thanks lovely person that gives me kudos
for 51^(25)/13