Find the remainder when $1021^{1022}$ is divided by $1023.$
(correct answer + 2, wrong answer 0)

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Solution

Note that $1024 = 2^{10} \equiv 1 (mod 1023)$
Then $1021^{1022} \equiv (- 2)^{1022}$
$\equiv (2)^{1022}$
$\equiv (2)^{10 \times 102 + 2}$
$\equiv 1024^{102} \times 2^{2}$
$\equiv 1^{102} \times 2^{2}$
$\equiv 4 (mod 1023)$

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Find the remainder when 10211022 is divided by 1023. (correct answer + 2, wrong answer 0)

Note that 1024=210≡1(mod 1023) Then 10211022≡(−2)1022 ≡(2)1022 ≡(2)10×102+2 ≡1024102×22 ≡1102×22 ≡4(mod 1023)

Find the remainder when $1021^{1022}$ is divided by $1023.$
(correct answer + 2, wrong answer 0)

Note that $1024 = 2^{10} \equiv 1 (mod 1023)$
Then $1021^{1022} \equiv (- 2)^{1022}$
$\equiv (2)^{1022}$
$\equiv (2)^{10 \times 102 + 2}$
$\equiv 1024^{102} \times 2^{2}$
$\equiv 1^{102} \times 2^{2}$
$\equiv 4 (mod 1023)$