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KVPY2020MathematicsBinomial Theorem

Let r be the remainder when 2021 2020 is divided by 2020 2 . Then r lies between

Options

  1. A0 and 5
  2. B10 and 15
  3. C20 and 100
  4. D107 and 120

Correct answer

A. 0 and 5

Step-by-step solution

( 2021 ) 2020 = ( 1 + 2020 ) 2020 C 0 2020 + C 1 2020 · 2020 + C 2 2020 · 2020 2 + … + C 2020 2020 · 2020 2020 1 + 2020 2 + C 2 2020 · 2020 2 + … + C 2020 2020 · 2020 2020 1 + 2020 2 1 + C 2 2020 + … + 2020 2018 1 + ( 2020 ) 2 · λ Hence ( 2021 ) 2020 = λ ( 2020 ) 2 + 1 Hence remainder = 1

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