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JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Binomial Theorem

The remainder when

(2021)^{2022} + (2022)^{2021}

is divided by 7 is

(A)0✓ Correct
(B)1
(C)2
(D)6

Answer: (A)

Solution

Write

2021 = 2023 - 2

and

2022 = 2023 - 1

, where

2023 = 7 \times 289

.

(2021)^{2022} + (2022)^{2021} = (2023 - 2)^{2022} + (2023 - 1)^{2021}

By binomial theorem,

(2023 - 2)^{2022} \equiv (- 2)^{2022} = 2^{2022} = 8^{674} = (7 + 1)^{674} \equiv 1 (mod 7)

Similarly,

(2023 - 1)^{2021} \equiv (- 1)^{2021} = - 1 (mod 7)

Therefore the sum

\equiv 1 + (- 1) = 0 (mod 7)

. Answer: (A)

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Single correct option

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