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JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Binomial Theorem

If

(2021)^{3762}

is divided by 17, then the remainder is ____

Answer: 4

Solution

Since

17 \times 119 = 2023

,

2021 \equiv - 2 (mod 17)

, so

202 1^{3762} \equiv (- 2)^{3762} = 2^{3762} (mod 17)

. Also

2^{4} = 16 \equiv - 1 (mod 17)

, hence

2^{8} \equiv 1 (mod 17)

. Writing

3762 = 8 \times 470 + 2

,

2^{3762} = (2^{8})^{470} \cdot 2^{2} \equiv 2^{2} = 4 (mod 17) .

The remainder is

4

.

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Integer answer

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