When I find GCF, I like to use the factor tree to do prime facotrization. The prime factorization of 10 is 2 and 5. The prime factorization of 12 is 2,2, and 3. The last step is to selected the prime numbers that are the same.
10: 2, 5
12: 2,2,3
As you can see, 2 is the only number that is in both of the prime factorization so the answer is 2.
