This question comes from the GMAT Prep Software
For any positive inter n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?
A. None
B. One
C. Two
D. Three
E. Four
I guessed on this question during the actual test.
While reviewing:
First, I tried multiplying the smallest prime number six times: 2*2*2*2*2*2=64 (that's 1 two-digit number with length of 6)
Then, I tried multiplying the second largest prime number six times: 3*3*3*3*3*3=81*9 which is not a two digit number. From here I knew that the 2 prime numbers had to be a product of mostly 2's and maybe a combination of 2 and another number.
Then I tried multiplying five 2's and one 3: 2*2*2*2*2*3=96 (this is the second two-digit number found with length 6)
Then I tried multiplying four 2's and two 3's: 2*2*2*2*3*3= 16*9= 108 (not a two digit number)
Because, by process of elimination, I found 2 two-digit numbers that satisfy the length of 6 rule. I determined that the answer is C (which is the correct answer)
This process felt a bit like guess work. Please give insight into other approaches to find this solution.
Thanks.
For any positive inter n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75=3*5*5. How many two-digit positive integers have length 6?
A. None
B. One
C. Two
D. Three
E. Four
I guessed on this question during the actual test.
While reviewing:
First, I tried multiplying the smallest prime number six times: 2*2*2*2*2*2=64 (that's 1 two-digit number with length of 6)
Then, I tried multiplying the second largest prime number six times: 3*3*3*3*3*3=81*9 which is not a two digit number. From here I knew that the 2 prime numbers had to be a product of mostly 2's and maybe a combination of 2 and another number.
Then I tried multiplying five 2's and one 3: 2*2*2*2*2*3=96 (this is the second two-digit number found with length 6)
Then I tried multiplying four 2's and two 3's: 2*2*2*2*3*3= 16*9= 108 (not a two digit number)
Because, by process of elimination, I found 2 two-digit numbers that satisfy the length of 6 rule. I determined that the answer is C (which is the correct answer)
This process felt a bit like guess work. Please give insight into other approaches to find this solution.
Thanks.
