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JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Binomial Theorem

The remainder when

3^{2022}

is divided by 5 is

(A)1
(B)2
(C)3
(D)4✓ Correct

Answer: (D)

Solution

We rewrite

3^{2022} = 9^{1011} = (10 - 1)^{1011}

. By the binomial theorem,

(10 - 1)^{1011} = \sum_{k = 0}^{1011} (k 1011) 1 0^{k} (- 1)^{1011 - k}

. All terms with

k \geq 1

are divisible by 10, hence by 5. The

k = 0

term is

(- 1)^{1011} = - 1

. Thus

3^{2022} \equiv - 1 \equiv 4 (mod 5)

. The remainder is 4.

JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Single correct option

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