Finding Remainder of Numbers of the Form a^b

Remainder whe...

Question

Remainder when $7^{636}$ is divided by 19 is equal to

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Solution

$7^{636} = (343)^{212} = (1 + 19 \times 18)^{212}$
$= 1 +^{212} C_{1} (19 \times 18) +^{212} C_{2} (19 \times 18)^{2} + . . . . . . .$
$\Rightarrow$ Remainder $= 1$

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