Question

what is the remainder when 7 ^2017 is divided by 25?

Answer 1

17 is the right answer

Answer 2

From binomial theorem, we know that

(a+b)^n=a^n+(nC1)ba^(n-1)+(nC2)(b^2)a^(n-2)+......+(nC(n-1))ab^(n-1)+b^n.

We can find the answer using this concept:

7^100=(7^2)^50=(49)^50

49^50=(50-1)^50

(50-1)^50=50^50+(50C1)*(-1)*50^49+(50C2)*{(-1)^2}*50^48+......+(50C49)*50*(-1)^49+(-1)^50

In this series, every term is multiplied at least by 50 except (-1)^50=1

Therefore the remainder obtained when 7^100 is divided by 25 is 1.