If n! = n*(n - 1)*(n - 2)*...*1, where n is a positive integer, which

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If \(n! = n*(n - 1)*(n - 2)*...*1\), where n is a positive integer, which of the following is (are) true?

I. 26 is not a factor of 25!
II. 29 is a factor of 25!
III. 1,000,000 is a factor of 25!

A. None
B. III Only
C. I and II only
D. II and III only
E. I, II, and III

I. 26 is not a factor of 25!

Since 26 = 2*13 and 25! contains both of these factors (25! = 1*2*...*13*...*25), then 26 IS a factor of 25!. Thus, the above statement is not true.

II. 29 is a factor of 25!

Since 29 is a prime number and 25! contains only the prime numbers up to 25, then 29 is NOT a factor of 25!. Thus, the above statement is not true.

III. 1,000,000 is a factor of 25!

The number of trailing zeros of 25! is calculated by 25!/5 + 25!/25 = 5 + 1 = 6 (check here for more: https://gmatclub.com/forum/everything-a ... 85592.html). Hence, 10^6 is indeed a factor of 25!. Thus, the above statement IS true.

Answer: B.

For more:

12. Trailing Zeros
13. Power of a number in a factorial

Hope this helps.­