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  4. The remainder when 2323 is divided by 53, is

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Q. The remainder when $23^{23}$ is divided by $53$, is

Binomial Theorem

Solution:

$23^{23} = 23 \times 23^{22} = 23 \times (23^2)^{11}$
$= 23 (529)^{11} = 23 (530 - 1)^{11}$
$= 23 (53\, m - 1)$, $m \in N$
$= 23 \times 53 \,m - 23 = 53 (23\, m - 1) + 30$
$\therefore $ Remainder is $30$.