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JEE Main202225 Jun 2022Morning ShiftMathematicsBinomial TheoremActual

If 1 2 · 3 10 + 1 2 2 · 3 9 + … + 1 2 10 · 3 = K 2 10 · 3 10 , then the remainder when K is divided by 6 is

Options

  1. A2
  2. B3
  3. C4
  4. D5

Correct answer

D. 5

Step-by-step solution

We know x n - y n = x - y x n - 1 + x n - 2 y + ⋯ x y n - 2 + y n - 1 1 2 · 3 10 + 1 2 2 · 3 9 + ⋯ · 1 2 9 · 3 2 + 1 2 10 · 3 = 2 9 + 2 8 · 3 +     ⋯     2 · 3 8 + 3 9 2 10 · 3 16 = 3 10 - 2 10 2 10 · 3 10 So, K = 3 10 - 2 10 Since we need the remainder when K is divided by 6 so, 3 10 = 6 q 1 + 3 and 2 10 = 6 q 2 + 4 Now K will be of the form 6 q 1 + 3 - 6 q 2 + 4 = 6 q 1 - q 2 - 1 Hence, when K is divided by 6 , we get the remainder

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