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.
