To find the greatest common factor (GCF) of the expressions [tex]\(8m\)[/tex], [tex]\(36m^3\)[/tex], and [tex]\(12\)[/tex], we need to examine both the numerical coefficients and the variable parts separately.
### Step 1: Finding the GCF of the coefficients
First, let's identify the numerical coefficients of each term:
- The coefficient of [tex]\(8m\)[/tex] is 8.
- The coefficient of [tex]\(36m^3\)[/tex] is 36.
- The coefficient of [tex]\(12\)[/tex] is 12.
We need to determine the GCF of these three numbers: 8, 36, and 12.
### Step 2: Prime factorization of the coefficients
Let's factorize each number into its prime factors:
- [tex]\(8 = 2 \times 2 \times 2 = 2^3\)[/tex]
- [tex]\(36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2\)[/tex]
- [tex]\(12 = 2 \times 2 \times 3 = 2^2 \times 3\)[/tex]
### Step 3: Finding the common factors
Now, we take the lowest powers of all common prime factors between them:
- The lowest power of 2 common to all three numbers is [tex]\(2^2\)[/tex], which is 4.
- The lowest power of 3 present is [tex]\(3^0\)[/tex], which is 1 since it is absent in 8 but present in 36 and 12.
Thus, the GCF of the coefficients 8, 36, and 12 is [tex]\(2^2 = 4\)[/tex].
### Step 4: Considering the variable part
Only two of the terms, [tex]\(8m\)[/tex] and [tex]\(36m^3\)[/tex], contain the variable [tex]\(m\)[/tex].
For the variable part:
- [tex]\(8m\)[/tex] has [tex]\(m^1\)[/tex].
- [tex]\(36m^3\)[/tex] has [tex]\(m^3\)[/tex].
- [tex]\(12\)[/tex] has no [tex]\(m\)[/tex] (which we can think of as [tex]\(m^0\)[/tex]).
Since [tex]\(m\)[/tex] must be present in the GCF between all three terms, and one of the terms [tex]\(12\)[/tex] has no [tex]\(m\)[/tex], the variable [tex]\(m\)[/tex] cannot be part of the GCF of all three terms.
### Step 5: Conclusion
Combining the results from the numerical part and the variable part, the greatest common factor of [tex]\(8m\)[/tex], [tex]\(36m^3\)[/tex], and [tex]\(12\)[/tex] is:
[tex]\[
\boxed{4}
\][/tex]
