Find the GCF (greatest common factor) of the following terms.
{90,30xy^2,60xy}
To find the greatest common factor (GCF) of the terms 90, 30xy^2, and 60xy, we need to consider the factors that are common to all terms.
First, we factorize each term into its prime factors:
- 90 can be factored as 2 × 3^2 × 5
- 30xy^2 can be factored as 2 × 3 × 5 × x × y^2
- 60xy can be factored as 2^2 × 3 × 5 × x × y
Next, we look for the common factors in all three expressions:
- The common prime factors are 2, 3, and 5
- The variable factors common to all terms are x and y (to the lowest powers present in all terms, which in this case is just y since y^2 is in the second term but not in the third).
Now, we take the lowest power of each common prime factor which is just 2, 3, and 5, and the common variable factors:
- The lowest power of 2 present in all terms is 2^1 (since 90 has a single 2 and is the smallest power of 2 present)
- The lowest power of 3 present in all terms is 3^1
- The lowest power of 5 present in all terms is 5^1
- The lowest power of x present in all terms is x^1 (since all terms include at least one x)
- The lowest power of y present in all terms is y^1 (since the third term only includes y to the first power)
The GCF is the product of these factors:
GCF = 2 × 3 × 5 × x × y = 30xy
So, the greatest common factor of 90, 30xy^2, and 60xy is 30xy.