25 25 is divided by 26 the remainder is

Asked by admin @ 23/08/2021 in Math viewed by 293 People

When 25∧25 is divided by 26 , the remainder is

Answered by admin @ 23/08/2021

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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