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JEE MainMathematicsBinomial Theorem

Let x denote the fractional part of x . If 23²⁰²³ + 30¹¹ 12 = R 12 , where R is a non-negative integer, then the value of R is

Options

  1. A1
  2. B11
  3. C5
  4. D7

Correct answer

B. 11

Step-by-step solution

The fractional part of a rational number N D is given by N D D . Thus, R is the remainder when 23²⁰²³ + 30¹¹ is divided by 12 . First, consider the term 23²⁰²³ modulo 12 : 23 = 24 - 1 = 12(2) - 1 23 -1 12 23²⁰²³ (-1)²⁰²³ -1 11 12 Next, consider the term 30¹¹ modulo 12 : 30 = 24 + 6 = 12(2) + 6 30 6 12 Notice that 6^2 = 36 0 12 . Therefore, for any integer n 2 , 6^n 0 12 . 30¹¹ 6¹¹ 0 12 Adding the two remainders: 23²⁰²³ + 30¹¹ 11 + 0 11 12 Hence, R = 11 . Answer: 11

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