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  4. When 2301 is divided by 5, the least positive remainder is

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Q. When $2^{301}$ is divided by $5$, the least positive remainder is

Principle of Mathematical Induction

Solution:

$2^4 \equiv 1 $(mod $5$) ;
$\Rightarrow (2^4)^{75} \equiv (1)^{75}$ (mod $5)$
i.e., $2^{300} \equiv 1 $(mod $5$)
$\Rightarrow 2^{300} \times 2 \equiv (1 \times 2) $( mod $5$)
$\Rightarrow 2^{301} \equiv 2($ mod $5$)
$\therefore $ Least positive remainder is $2$.