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JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Binomial Theorem

The remainder when

(2023)^{2023}

is divided by 35 is ______.

Answer: 7

Solution

Since

2023 = 35 \times 58 - 7

, we have

2023 \equiv - 7 (mod 35)

, so

202 3^{2023} \equiv (- 7)^{2023} = - 7^{2023} (mod 35)

. The powers of

7

modulo

35

cycle as

7, 14, 28, 21

with period four, and

2023 \equiv 3 (mod 4)

, so

7^{2023} \equiv 28 (mod 35)

. Hence

202 3^{2023} \equiv - 28 \equiv 7 (mod 35)

, and the remainder is

7

.

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Integer answer

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