Multiple choice

The highest power of $2$ by which the product of first $100$ counting numbers can be divided without any remainder is $.......$

  1. $97$
  2. $96$
  3. $95$
  4. $94$
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A Correct answer
Explanation

The highest power of a prime p dividing n! is given by Legendre's formula: sum of floor(n/p^k). For 100! and p=2, this is floor(100/2) + floor(100/4) + floor(100/8) + floor(100/16) + floor(100/32) + floor(100/64) = 50 + 25 + 12 + 6 + 3 + 1 = 97.

AI explanation

To find the highest power of 2 that divides 100 factorial, use Legendre's formula to sum the integer quotients of 100 divided by powers of 2. Dividing 100 by 2 gives 50, dividing 100 by 4 gives 25, dividing by 8 gives 12, dividing by 16 gives 6, dividing by 32 gives 3, and dividing by 64 gives 1. Adding these quotients, 50 plus 25 plus 12 plus 6 plus 3 plus 1, results in 97.

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