The product of the last two digits of (1919)^ 1919 is __________ - Vidyadip

Question

The product of the last two digits of $(1919)^{1919}$ is __________

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Answer

(63)
$(1919)^{1919}=(1920-1)^{1919}$
$={ }^{1919} \mathrm{C}_{0}(1920)^{1919}-{ }^{1919} \mathrm{C}_{1}(1920){ }^{1918}+\ldots .$.
$+{ }^{1919} \mathrm{C}_{1918}(1920){ }^{1}-{ }^{1919} \mathrm{C}_{1919}$
$=100 \lambda+1919 \times 1920-1$
$=100 \lambda+3684480-1$
$=100 \lambda+\ldots \ldots \ldots . .79$ (last two digit)
$\Rightarrow$ Number having last two digit 79
$\therefore$ Product of last two digit 63

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