enigma123 wrote:
If d is a positive integer and f is the product of the first 30 positive integers, what is the value of d?
(1) 10^d is a factor of f
(2) d>6
We are given that d is a positive integer and f = 30!. We need to determine the value of d.
Statement One Alone:
10^d is a factor of f
Since 10^1 and 10^2 could each divide into 30!, we do not have a unique value for d. Statement one alone is not sufficient to answer the question.
Statement Two Alone:
d > 6
Since d could be 7, 8, or greater, statement two alone does not allow us to determine a unique value of d.
Statements One and Two Together:
Using both statements, since we know that d > 6, let’s determine the maximum value d can be given that 10^d divides into 30!. Essentially, we need to determine the maximum number of five-two pairs. (Recall that each five-two pair creates a factor of 10.) Since there are more twos than fives, let’s determine the number of fives.
The factors that are multiples of 5 in 30! are 5, 10, 15, 20, 25 = 5^2, and 30. So, we see there are 7 fives in 30!, and thus the maximum value of d is 7. Since d > 6, d must be 7.
Answer: C
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