Question

if 25^25 is divided by 26 then what will be the remainder?

Answer 1

Answer: 25

Step-by-step explanation:

We can write that

2525=(26−1)25.

Now, by the binomial theorem, on expanding the above expression, we will get twenty-six terms.

(26–1)25=25C02625×10−25C12624×11+…+25C24261×124−25C25260×125

Twenty-five of these twenty-six terms will have 26 as a factor. So, these twenty-five terms do not contribute anything to the remainder. Finally we have one term left. It is -1.

Hence the remainder should be -1. But since generally we take into account positive numbers as remainders, so the remainder will be 26+(−1)=25.

Hence the remainder is 25.

You can easily see the when you divide 25x (where x is an odd natural number) by 26 the remainder is 25, and the remainder is 1 if x is an even natural number.