The remainder when $2^{2016}$ is divided by $63$, is
- 1
- 8
- 17
- 32
The Correct Option is A
Solution and Explanation
\(2^{2016}=\left(2^{6}\right)^{336}=(64)^{336}=(63+1)^{336}\)
\((63+1)^{336}=^{366}C_0(63)^0(1)^{366}+^{366}C_1(63)^1(1)^{364}+^{366}C_2(63)^2(1)^{362}+…… = 1+63K\)
\(\therefore\) Remainder \(=1\)
So, the correct option is (A): \(1\)
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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.