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JEE Main · 2022 · 28 Jul 2022 · Shift 1 · Binomial Theorem

The remainder when

7^{2022} + 3^{2022}

is divided by 5 is:

(A)0
(B)2
(C)3✓ Correct
(D)4

Answer: (C)

Solution

We use modular arithmetic and binomial expansion. Note that

7 \equiv 2 (mod 5)

and

3 \equiv 3 (mod 5)

.

Write

7^{2022} = (49)^{1011} = (50 - 1)^{1011} \equiv (- 1)^{1011} = - 1 (mod 5)

.

Similarly,

3^{2022} = (9)^{1011} = (10 - 1)^{1011} \equiv (- 1)^{1011} = - 1 (mod 5)

.

Therefore,

7^{2022} + 3^{2022} \equiv - 1 + (- 1) = - 2 \equiv 3 (mod 5)

.

The remainder is 3.

JEE Main · 2022 · 28 Jul 2022 · Shift 1 · Single correct option

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