M7MBA wrote:If k is a positive integer and m is the product of the first 40 positive integers, what is the value of k ?
(1) 10^k is a factor of m.
(2) 10^k is a factor of n, where n is the product of the first 9 positive integers.
The OA is the option B.
Using the statement (2), I don't need to use m? I am confused here. Experts, may you help me?
Statement 1: Case 1: k = 1. I0^1 is.a factor of 1*2*3...40, because 1*2*3...40/(10^1) = integer. Put another way, the product contains at least one 10, and thus 10^1 is a factor of the product.
Case 2: k =2. 10^2 is a factor of 1*2*3...40, because 1*2*3...40/(10^2) = integer. Put another way, the product contains at least two 10's, and thus 10^2 is a factor of the product.
Because k can assume more than one value, statement 1 is not sufficient.
Statement 2: Case 1: k = 1. 10^1 is a factor of 1*2*3...9 because 1*2*3...9/(10^1) = integer. (The 10 would cancel out the 2 and the 5 in the numerator. You can visualize it like this: 1*
2*3*4*
5*6*7*8*9/
2*5) The red terms cancel out, leaving you with an integer.
No other case will work if k must be positive. 10^2 is NOT a factor of 1*2*3...9 because 1*2*3...9/(10^2) is NOT an integer. Play the same game: 1*
2*3*4*
5*6*7*8*9/
2*5*2*5) The red terms cancel out, leaving you with 1*3*4*6*7*8*9/2*5. Well, you could cancel out the 2 in the denominator, but that 5 isn't going anywhere, thus we're left with a fraction, and so 10^2 is NOT a factor of the product, and k cannot be 2. (Or any greater integer.) Statement 2 alone is sufficient. The answer is
B
