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Q. The last two digits of the number $\left(23\right)^{14}$ are

NTA AbhyasNTA Abhyas 2022

Solution:

$(23)^{14}=(529)^{7}=(530-1)^{7}$
$={ }^{7} C_{0}(530)^{7}-{ }^{7} C_{1}(530)^{6}+\ldots \ldots .-{ }^{7} C_{5}(530)^{2}+{ }^{7} C_{6}(530)-1$
$=100 K+7 \times 530-1$
$=100 K+3709$
$\Rightarrow$ last two digits are 09