The remainder when 428 2024 is divided by 21 is

Question

The remainder when $428^{2024}$ is divided by 21 is _______.

Smart Achievers · Accepted Answer

(1)

$We have,$ $428^{2024} = {(420 + 8)}^{2024}$

$= {}^{2024}C_{0} {(420)}^{2024} + {}^{2024}C_{1} {(420)}^{2023} {(8)}^{1} + \dots + {}^{2024}C_{2023} (420) 8^{2023} + {}^{2024}C_{2024} 8^{2024}$

$= 420 [Some integer] + 64^{1012}$

$= 21 \times 20 [Some integer] + {(63 + 1)}^{1012}$

$= 21 \times 20 [Some integer] + 63 \times Some integer + 1$

$= 21 \times 20 [Some integer] + 21 \times 3 \times Some integer + 1$

$Hence, remainder = 1$

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