if d is a positive integer and f is the product

[john_haddock]

if d is a positive integer and f is the product of the first 30 integers what is the value of d?

(1) 10^d is a factor of f
(2) d>6

[stock.mojo11]

if d is a positive integer and f is the product of the first 30 integers what is the value of d?

(1) 10^d is a factor of f
(2) d>6

Did you mean first 30 positive integers? Assuming thats what you meant, the answer is C

d > 6 Insuff

10 ^ d is a factor of f 10 is a factor and 10 ^ 2 is a factor too. Hence d = 1/2 or more Insuff

Combine. How many 10's are in 30!

3 10's (10,20,30)

Now 10 can also be 5 X 2. rather than counting 2's and 5's just count 5's as there can be only that many number of 10's

5,15,25 ( total of 4 5's) Note that there are two 5's in 25.

Hence the max number of 10's in 30! is 7. means 10 ^ 7 is a factor of 30! but d > 6 so d =7

[RonPurewal]

Combine. How many 10's are in 30!

3 10's (10,20,30)

Now 10 can also be 5 X 2. rather than counting 2's and 5's just count 5's as there can be only that many number of 10's

5,15,25 ( total of 4 5's) Note that there are two 5's in 25.

this solution is good, in that it gets at the underlying structure of the problem (i.e., this problem is ultimately about prime factorizations).

however, it's unnecessarily try-hard.

here's all you have to do:
forget entirely about 10, 20, and 30, and ONLY THINK ABOUT PRIME FACTORIZATIONS.
(TAKEAWAY: this is the way to go in general - when you break something down into primes, you should not think in hybrid terms like this. instead, just translate everything into the language of primes.)

each PAIR OF A '5' AND A '2' in the prime factorization translates into a '10'.

there are seven 5's: one each from 5, 10, 15, 20, and 30, and two from 25.

there are waaaaaaayyyyy more than seven 2's.

therefore, 30! can accommodate as many as seven 10's before you run out of fives.

--

statement 2 is clearly insufficient.

statement 1, by itself, means that d can be anything from 1 to 7 inclusive.

together, d must be 7.

ans (c)