To determine the greatest number of portions of trail mix that Keenan can make, we need to find the greatest common divisor (GCD) of the number of bags of nuts and the number of bags of dried fruit. This will give us the maximum number of identical portions that can be made without any bags left over.
Keenan has:
- 18 bags of nuts
- 12 bags of dried fruit
We will calculate the GCD of 18 and 12.
First, we can find the prime factorizations of each number:
- The prime factorization of \( 18 \) is: \[ 18 = 2 \times 3^2 \]
- The prime factorization of \( 12 \) is: \[ 12 = 2^2 \times 3 \]
Next, we find the GCD by identifying the lowest power of each prime factor present in both factorizations:
- For the prime number \( 2 \), the minimum power is \( 2^1 \) (from \( 18 \)).
- For the prime number \( 3 \), the minimum power is \( 3^1 \) (from \( 12 \)).
Now we can multiply these together to obtain the GCD: \[ \text{GCD} = 2^1 \times 3^1 = 2 \times 3 = 6 \]
Thus, the greatest number of identical portions of trail mix that Keenan can create is \( \boxed{6} \).