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COMEDK2022MathematicsBinomial Theorem

If 4^n+15 n+P is divisible by 9 for all n N , then the least negative integral value of P is

Options

  1. A-1
  2. B-2
  3. C-3
  4. D-4

Correct answer

A. -1

Step-by-step solution

Let f(n) = 4^n + 15n + P . We are given that f(n) is divisible by 9 for all n N . For n=1 , f(1) = 4^1 + 15(1) + P = 19 + P . Since f(1) is divisible by 9, 19 + P = 9k for some integer k . For n=2 , f(2) = 4^2 + 15(2) + P = 16 + 30 + P = 46 + P . Since f(2) is divisible by 9, 46 + P = 9m for some integer m . Subtracting the two expressions: (46 + P) - (19 + P) = 27 , which is divisible by 9. This confirms the condition is consistent for n=1 and n=2 . From 19 + P = 9k , we have P = 9k - 19 . To find the least negati

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