The LCM and GCF of three numbers are 180 and 12 respectively. Two numbers are 36 and 60. Find the third possible number?

L.C.M. = 180 = 2² × 3² × 5
G.C.F. = 12 = 2² × 3

1st number = 36 = 2² × 3²
2nd number = 60 = 2² × 3 × 5

Consider the prime factor 2, the third number must have a factor of 2².
Consider the prime factor 3, the third number has a factor of 3 or 3².
Consider the prime factor 5, the third number may have a factor 5 (or may not).

The possible third number:
2² × 3 = 12
2² × 3² × 5 = 180

Prime factors of 36 are 2, 3. Prime factorization of 36 in exponential form is:

36 = 2^2 × 3^2

Prime factors of 60 are 2, 3, 5. Prime factorization of 60 in exponential form is:

60 = 2^2 × 3^1 × 5^1

Factors of 36
List of positive integer factors of 36 that divides 36 without a remainder.

1, 2, 3, 4, 6, 9, 12, 18

Factors of 60
List of positive integer factors of 60 that divides 36 without a remainder.

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30

Greatest Common Factor
We found the factors and prime factorization of 36 and 60. The biggest common factor number is the GCF number.
So the greatest common factor 36 and 60 is 12.