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NTA Abhyas JEE Main2020MathematicsBinomial TheoremPractice

If 5 97 is divided by 52 , then the remainder obtained is

Options

  1. A3
  2. B5
  3. C4
  4. D0

Correct answer

B. 5

Step-by-step solution

We know that, 5 4 = 625 = 52 × 12 + 1 ⇒ 5 4 = 52 λ + 1 , where λ is a positive integer. ⇒ 5 4 24 = 52 λ + 1 24 = C 0 24 52 λ 24 + C 1 52 λ 23 + C 2 24 52 λ 22 24 + … + C 23 52 λ + C 24 24 24 (by binomial theorem) ⇒ 5 96 = 52 C 0 24 52 23 λ 24 + C 1 52 23 λ 22 24 + … + C 23 λ 24 + 1 = a m u l t i p l e o f 52 + 1 On multiplying both sides by 5 , we get 5 97 = 5 96 ⋅ 5 = 5 a m u l t i p l e o f 52 + 5 Hence, the required remainder is 5 .

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