LCM of X and 12?

[milanproda]

What is the LCM of X and 12?

(1)- The LCM of X and 9 is 45
(2)- The LCM of X and 4 is 20

I am confused as to how to answer this q.

Thanks!

[SoCan]

Prime factors of 12 are 2^2 and 3, so we want to know if x has any different prime factors, or more 2s or more 3s.

1) The prime factors of 45 are 3^2 and 5. The prime factor breakdown for 9 is 3^2. So what are the prime factors of x? We know it must contain one 5, since 45 contains one 5 and 9 doesn't contain any. However, we don't know if it contains zero 3s, one 3 or two 3s. The LCM is the same whether x has zero or one 3s (since 12 already has one 3), but it's different if x has two 3s. Insufficient.

2) The prime factors of 20 are 2^2 and 5. The prime factor breakdown for 4 is 2^2. What are the prime factors of x? We know it must contain one 5, since 20 contains one 5 and 4 doesn't contain any. While we don't know how many 2s x contains, it doesn't matter because the most it can have is two, which is the same number of 2s that 12 has. i.e., it doesn't matter if x is equal to 5, 10 or 20, the LCM between x and 12 is still 60. Sufficient.

[cans]

a) lcm of X and 9 is 45
X can be 5 or 15 or 45
if X is 5 or 15, lcm of X and 12 is 60 but if X is 45, lcm is 120.
Insufficient
b)lcm of x and 4 is 20
X can be 5,10,20
if X is 5 or 10, 20 lcm of x and 12 is 60
Sufficient
IMO B