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JEE MainMathematicsBinomial Theorem

The fractional part of the number 5⁹⁹ 13 is

Options

  1. A5 13
  2. B12 13
  3. C8 13
  4. D7 13

Correct answer

C. 8 13

Step-by-step solution

5⁹⁹ 13 = 5 (5^2)⁴⁹ 13 = 5 (25)⁴⁹ 13 We can write 25 as 26 - 1 . 5 (26 - 1)⁴⁹ 13 = 5 (⁴⁹C₀ 26⁴⁹ - ⁴⁹C₁ 26⁴⁸ + - ⁴⁹C₄₉ 1⁴⁹ ) 13 Let the terms containing 26 be 26k , where k is an integer. = 5 (26k - 1) 13 = 130k - 5 13 = 10k - 5 13 To find the positive fractional part, we rewrite the expression as: 10k - 1 + 1 - 5 13 = (10k - 1) + 8 13 Since 10k - 1 is an integer, the fractional part is 8 13 . Answer: 8 13

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