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COMEDK2022MathematicsBinomial Theorem

(2^ 3 n -1 ) is divisible by

Options

  1. A6
  2. B7
  3. C8
  4. D9

Correct answer

B. 7

Step-by-step solution

The expression is 2^ 3n - 1 . We can rewrite 2^ 3n as (2^3)^n . Since 2^3 = 8 , the expression becomes 8^n - 1 . Using the algebraic identity x^n - 1 = (x - 1)(x^ n-1 + x^ n-2 + + 1) , we substitute x = 8 . 8^n - 1 = (8 - 1)(8^ n-1 + 8^ n-2 + + 1) = 7(8^ n-1 + 8^ n-2 + + 1) . This shows that the expression is always divisible by 7 for any positive integer n . Answer: 7

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