Question:

The remainder when 220162^{2016} is divided by 6363, is

Updated On: Aug 2, 2024
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The Correct Option is A

Solution and Explanation

22016=(26)336=(64)336=(63+1)3362^{2016}=\left(2^{6}\right)^{336}=(64)^{336}=(63+1)^{336}

(63+1)336=366C0(63)0(1)366+366C1(63)1(1)364+366C2(63)2(1)362+……=1+63K(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=1

So, the correct option is (A): 11

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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.