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# If $25^{15}$ is divided by 13 the remainder is ___________

$\begin{array}{1 1}(A)\;10\\(B)\;12\\(C)\;14\\(D)\;16\end{array}$

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## 1 Answer

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Toolbox:
• $(x-a)^n=nC_0x^n a^0-nC_1x^{n-1} a^1+nC_2 x^na^2.......nC_n x^0 a^n$
$(25)^{15}=(26-1)^{15}$
$\Rightarrow 15C_0 (26)^{15}(-1)^0+15C_2 (26)^{14}(-1)^1+15C_{15}.(26) (-1)^{15}$
Hence the last term $2b=13\times 2$ which is divisible by 13.Hence no remainder.
$\Rightarrow$ The last term is $(-1)^{15}=-1$
Hence the remainder of (-1) by 13 is $-1=13\times (-1)+12$
Hence the answer is 12.
answered Jun 25, 2014

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