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JEE Main20236 Apr 2023Evening ShiftMathematicsBinomial TheoremActual

Among the statements : ( S 1 ) : 2023 2022 - 1999 2022 is divisible by 8 . ( S 2 ) : 13 ( 13 ) n - 11 n - 13 is divisible by 144 for infinitely many n ∈ ℕ

Options

  1. AOnly S 2 is correct
  2. BOnly ( S 1 ) is correct
  3. CBoth S 1 and S 2 are correct
  4. DBoth S 1 and S 2 are incorrect

Correct answer

B. Only ( S 1 ) is correct

Step-by-step solution

Given, ( S 1 )   :   ( 2023 ) 2022 - ( 1999 ) 2022 is divisible by 8 Now we know that ( x - y ) divides x n - y n   ∀ n ∈ ℕ So, 2023 - 1999 divides ( 2023 ) 2022 - ( 1999 ) 2022 ⇒   24 divides ( 2023 ) 2022 - ( 1999 ) 2002 ⇒   8 will divide ( 2023 ) 2022 - ( 1999 ) 2002 As 8 divides 24 Hence, S 1 is correct Now solving, S 2   :   13 13 n - 11 n - 13 is divisible by 144 for n ∈ ℕ So using binomial theorem in 1 + 12 n we get, 13 1 + 12 n - 1

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