a prime factorization tough DS problem

[wind]

How much is the LCM of x and 12 ?
1-LCM x and 9 = 45
2-LCM x and 4 = 20

Ron gave this DS problem in the end of the June the 10th , 2010 studyhall . Running out of time , he mentioned the correct answer is B , but I tried it many times and it is always D . Any advice ?

[nt2011]

Per Stmt 1, x can be 5, 45, 15 and this results LCM of x and 12 as 180, 60 so Not Sufficient

Per Stmt 2, x can be 5, 10, 20 and this results in LCM of x and 12 as always 60 so Sufficient.

Hence B is the answer.

[jlucero]

Great explanation nt2011, and exactly how I'd do this problem... finding the largest and smallest (and then double checking all) possible values of x.

The reason for the difference comes from the 12. 12 = 2x2x3. Note this has two 2s and one 3.

1) x must be divisible by 5 and is divisible by zero, one, or two 3s.
2) x must be divisible by 5 and is divisible by zero, one, or two 2s.

In the first scenario, being divisible by zero 3s or two 3s means a difference in the LCM (hence 60 or 180)

In the second scenario, 12 is already divisible by zero, one, or two 2s, so the LCM of x and 12 will be the same- 60.