Multiple choice general knowledge math & puzzles

what is the maximum value of n such that 77! is perfectly divisible by 720^n(720 reaise to power n)?

  1. 35

  2. 18

  3. 17

  4. 36
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C Correct answer
Explanation

To find the maximum n such that 77! is divisible by 720^n, we factor 720 = 2^4 × 3^2 × 5. In 77!, the exponent of 2 is 73, of 3 is 35, and of 5 is 18. For 720^n = 2^(4n) × 3^(2n) × 5^n, we need 4n ≤ 73, 2n ≤ 35, and n ≤ 18. The limiting factor is 3, giving n ≤ 17 (since 35/2 = 17.5). Thus, maximum n = 17.

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