Here is one elementary way of finding the greatest common factor, by finding the prime factorization of each number.
36 = 2*2*3*3
48 = 2*2*2*2*3
60 = 2*2*3*5
The greatest common factor is the product of all the primes that are common to all three numbers: 2*2*3 = 12.
And here is one of many other ways of finding the greatest common factor.
36 48 60
|
+--------------
Find any factor common to all three numbers; write that number on the left, and remove that common factor from each of the three numbers:
36 48 60
3 | 12 16 20
+--------------
Continue doing that until there are no more common factors in the three remaining numbers:
36 48 60
3 | 12 16 20
2 | 6 8 10
2 | 3 4 5
+--------------
The greatest common factor of the three numbers is the product of the common factors you have removed: 3*2*2 = 12.
Note this process also finds the least common multiple of the three numbers. It is the greatest common factor, times the numbers that remain: (3*2*2)*(3*4*5) = 12*60 = 720.