Ryan's error is that he did not include only common prime factors in the greatest common factor.
The prime factorization for 48 is \(2^4 \times 3^1\) (or \(2 \times 2 \times 2 \times 2 \times 3\)), and for 72 it is \(2^3 \times 3^2\) (or \(2 \times 2 \times 2 \times 3 \times 3\)).
To find the greatest common factor (GCF), we take the lowest powers of all the common prime factors. In this case, the common prime factors are 2 and 3:
- For 2, the minimum power is \(2^3\) (from 72).
- For 3, the minimum power is \(3^1\) (from 48).
Therefore, the GCF is:
\[ 2^3 \times 3^1 = 8 \times 3 = 24 \]
Ryan incorrectly included \(3^2\) in his calculation, which is why the greatest common factor he found was incorrect.