Answers

  • Brainly User
2015-01-08T21:44:13+05:30

Method (1):
Consider, f(x)=x^{25}+1.
As you can see f(-1)=0 which means (x+1) is a factor of f(x).
Hence, f(x)=(x+1)g(x) for some g(x) of degree 24.
Now, 25^{25}=(25^{25}+1)-1=f(25)-1=(25+1)g(25)-1=26(g(25)-1)+25.
Hence, remainder is 25.

Method (2):
From Binomial Theorem,
25^{25}=(26-1)^{25}=^{25}C_{0}(26)^{25}+^{25}C_{1}(26)^{24}(-1)^{1}+.....+^{25}C_{25}(-1)^{25} .
So the last term is "-1". Hence remainder is "26+(-1)" = 25.

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