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AP EAMCET202422 May 2024Evening ShiftMathematicsBinomial TheoremActual

If P is the greatest divisor of 49^ n +16 n-1 for all n ~N , then the number of factors of P is

Options

  1. A12
  2. B15
  3. C7
  4. D13

Correct answer

C. 7

Step-by-step solution

aligned & 49^n+16 n-1=(1+48)^n+16 n-1 = & 1+ ^n C₁(48)+ ^n C₂(48)^2+ .+ ^n C_n(48)^n+16 n-1 = & (48 n+16 n)+ ^n C₂(48)^2+ .+ ^n C_n(48)^n = & 64 n+8^n ( ^n C₂ 6^2+ ^n C₃ 6^3 8+ + ^n C_n 6^n 8^ n-2 ) aligned Hence, 49^ n +16 n-1 is divisible by 64 and factors of 64 are 1,2,4,8,16,32,64 . So, number of factors =7

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