If you break down each of these numbers into their prime factorizations (one way to do this is with a factor tree), we have 68 = 2 x 2 x 17 = 2^2 x 17, and 24 = 2 x 2 x 2 x 3 = 2^3 x 3.

68 has two distinct factors, 2 and 17. 24 has two distinct factors, 2 and 3. A common factor is a number that both of our original numbers can be divided by without resulting in a remainder. The GCF is even more specific...

[Compare each factor that our original numbers have in common. The one with the smallest exponent will become part of our GCF. Continue to compare each factor that they share until all common factors have been compared. Multiply these numbers together and you have the overall GCF.]

...So, the only factor that is common between 68 and 24 is the factor 2. It is raised to the 2nd power in the factorization of 68, and to the 3rd power in the factorization of 24. We can only divide by the smaller of the two factors, 2^2 by BOTH 68 and 24 so 2^2, or 4, is the GCF.

68 has two distinct factors, 2 and 17. 24 has two distinct factors, 2 and 3. A common factor is a number that both of our original numbers can be divided by without resulting in a remainder. The GCF is even more specific...

[Compare each factor that our original numbers have in common. The one with the smallest exponent will become part of our GCF. Continue to compare each factor that they share until all common factors have been compared. Multiply these numbers together and you have the overall GCF.]

...So, the only factor that is common between 68 and 24 is the factor 2. It is raised to the 2nd power in the factorization of 68, and to the 3rd power in the factorization of 24. We can only divide by the smaller of the two factors, 2^2 by BOTH 68 and 24 so 2^2, or 4, is the GCF.