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Q.
The last two digits of the number $(23)^{14}$ are
Binomial Theorem
Solution:
$(23)^{14}=(529)^{7}=(530-1)^{7} $
$={ }^{7} C _{0}(530)^{7}-{ }^{7} C _{1}(530)^{6}+\ldots-{ }^{7} C _{5}(530)^{2}+{ }^{7} C _{6} 530-1 $
$={ }^{7} C _{0}(530)^{7}-{ }^{7} C _{1}(530)^{6}+\ldots .+3710-1$
$=100 \,m +3709$
Therefore, last two digits are $09 $.