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99 Percentile Qs Bank for JEE MainMathematicsBinomial Theorem

The greatest integer which divides the number 101 100   - 1 is

Options

  1. A100
  2. B1000
  3. C10000
  4. D100000

Correct answer

C. 10000

Step-by-step solution

For the greatest integer which divides the number 101 100 - 1, consider 1 + x n = 1 + nx + n n - 1 2 x 2 + ........ + x n by Binomial theorem) ⇒ 1 + x n - 1 = nx + n - 1 2 x 2 + ........ + x n If x   =   n , then; 1 + n n - 1 = n 2 + n n - 1 2 n 2 + ... + n n ⇒ 1 + n n - 1 = n 2 1 + n n - 1 2 + ... n n - 2 Put n = 100 in the equation, 1 + 1 0 0 1 0 0 - 1 = 1 0 0 2 1 + 1 0 0 1 0 0 - 1 2 + ........ + 1 0 0 9 8 ⇒ 1 0 1 1 0 0 - 1 = 1 0 0 2 1 + 1 0 0 × 9 9 2 + ... + 1 0 0 9 8 ⇒ 101

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