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.
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Answers & Comments
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.