You really don't need prime factorization to find the LCM of two numbers. LCM stands for Least Common Multiple. So, write down the multiples of each number until you find one that is "common" to both. That will be the LCM.
Multiples of 9 : 9, 18, 27, 36, 45, 54, 63
Multiples of 21: 21, 42, 63
Therefore 63 is the LCM.
Using prime factorization,
9 factors into 3 x 3
21 factors into 3 x 7
Multiply each factor the most # of times it occurs in each number.
The LCM would be 3 x 3 x 7