The remainder when (2021)2023 is divided by 7 is
1
2
5
6
The Correct Option is C
Solution and Explanation
The correct answer is (C) : 5
2021 ≡ –2 (mod 7)
⇒ (2021)2023 ≡–(2)2023 (mod 7)
≡ –2(8)674 (mod 7)
≡ –2(1)674 (mod 7)
≡ –2(mod 7)
≡ 5(mod 7)
So when (2021)2023 is divided by 7, remainder is 5.
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.