The correct options are iii. 54p^(3) and iv. 63p^(3)q^(6
To determine which terms have a greatest common factor (GCF) of 6p^(3), we need to look for factors that are common to all the terms.
First, let's break down the terms into their prime factors:
i. 12p^(3)r = 2 * 2 * 3 * p * p * p * r
ii. 27p^(4)q = 3 * 3 * 3 * p * p * p * p * q
iii. 54p^(3) = 2 * 3 * 3 * 3 * p * p * p
iv. 63p^(3)q^(6) = 3 * 3 * 7 * p * p * p * q * q * q * q * q * q
Now, let's identify the common factors among the terms:
i. 12p^(3)r = 2 * 2 * 3 * p * p * p * r
ii. 27p^(4)q = 3 * 3 * 3 * p * p * p * p * q
iii. 54p^(3) = 2 * 3 * 3 * 3 * p * p * p
iv. 63p^(3)q^(6) = 3 * 3 * 7 * p * p * p * q * q * q * q * q * q
From the prime factorization, we can see that the common factors are 2, 3, and p^(3).
Among the given options, only iii. 54p^(3) and iv. 63p^(3)q^(6) have a GCF of 6p^(3). The other options either don't have a factor of 6 or don't have a factor of p^(3).