Find the greatest common factor of 27p^12q^2, 63p^3q^6, and 108p^2q^8

Answer 1

Notice that each of these terms can be broken in a product of terms, namely:

27 *p^12 *q^2

63 *p^3 *q^6

108 *p^2 *q^8

Next, a helpful trick is that:

GCF(27p^12q^12, 63p^3q^6, 108p^2q^8) = GCF(27,63,108) * GCF(p^12,p^3,p^2) * GCF(q^2,q^6,q^8).

First, to find GCF(27,63,108), we only have to look at the largest factor which divides both the smallest term as well as the other terms. This makes sense, since if that factor does not divide the smallest term, it cannot be the GCF. Notice that the factors of 27 are 1,3,9, and 27. Well, 27 does not divide 63, so it cannot be the GCF. But the next largest factor, 9, divides all three: 9*7 = 63 and 9*12 = 108. And so 9 is the GCF of these terms.

Next, finding the GCF of the other two terms is a bit easier. For GCF(p^12,p^3,p^2), notice that p^3/p^2 = p and p^12/p^2 = p^10. So, since the smallest term divides the two other terms, p^2 is the GCF for this set of terms.

Finally, for GCF(q^2,q^6, q^8), notice that q^6/q^2 = q^4 and q^8/q^2 = q^6. And so again, we have the smallest term dividing the two others. This means that q^2 is the GCF for this set of terms.

Putting this all together, we have:

GCF(27p^12q^12, 63p^3q^6, 108p^2q^8)

= GCF(27,63,108) * GCF(p^12,p^3,p^2) * GCF(q^2,q^6,q^8).

= 9* p^2 * q^2.

Try dividing each of your terms by a larger expression and notice that they will result in a fraction.