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  4. The least remainder when 1730 is divided by 5 , is

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Q. The least remainder when $17^{30}$ is divided by $5$ , is

ManipalManipal 2011

Solution:

$(17)^{2}=289 \equiv(-1)(\bmod 5)$
$\Rightarrow \left(17^{2}\right)^{15} \equiv(-1)^{15}(\bmod 5)$
$\Rightarrow 17^{30} \equiv-(\bmod 5)$
i.e., $17^{30} \equiv 4(\bmod 5)$