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JEE Main202310 Apr 2023Evening ShiftMathematicsBinomial TheoremActual

Let the number ( 22 ) 2022 + ( 2022 ) 22 leave the remainder α when divided by 3 and β when divided by 7 . Then ( α 2 + β 2 ) is equal to

Options

  1. A20
  2. B13
  3. C5
  4. D10

Correct answer

C. 5

Step-by-step solution

Given that α be the remainder when 22 2022 + 2022 22 is divided by 3 and β be the remainder when the same is divided by 7 . ⇒ 22 2022 + 2022 22 = 21 + 1 2022 + 2022 22 . Here 2022 22 is divisible by 3 as 2022 is divisible by 3 . So on expanding 21 + 1 2022 , we get ⇒ 21 + 1 2022 = C 0 2022 21 2022 + C 1 2022 21 2021 + . . . . . . + C 2022 2022 1 2022 = 3 3 2021 × 7 2022 + C 1 2022 × 3 2020 × 7 2021 + . . . . . . + 1 = 3 k 1 + 1 In this case the remainder is 1 Hence, α = 1 N

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