To find the Greatest Common Factor (GCF) of the three monomials [tex]\(27x^4y^3\)[/tex], [tex]\(12x^2y\)[/tex], and [tex]\(51xy^2\)[/tex], we need to consider both the coefficients and the variable parts (i.e., exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]).
### Step 1: Find the GCF of the coefficients
The coefficients are [tex]\(27\)[/tex], [tex]\(12\)[/tex], and [tex]\(51\)[/tex].
- The prime factorization of [tex]\(27\)[/tex] is [tex]\(3^3\)[/tex].
- The prime factorization of [tex]\(12\)[/tex] is [tex]\(2^2 \cdot 3\)[/tex].
- The prime factorization of [tex]\(51\)[/tex] is [tex]\(3 \cdot 17\)[/tex].
The common prime factor is [tex]\(3\)[/tex], and the smallest power of [tex]\(3\)[/tex] among the coefficients is [tex]\(3^1\)[/tex].
Thus, the GCF of the coefficients is [tex]\(3\)[/tex].
### Step 2: Find the GCF of the variable parts
We consider each variable separately.
#### For [tex]\(x\)[/tex]:
- In [tex]\(27x^4y^3\)[/tex], the exponent of [tex]\(x\)[/tex] is [tex]\(4\)[/tex].
- In [tex]\(12x^2y\)[/tex], the exponent of [tex]\(x\)[/tex] is [tex]\(2\)[/tex].
- In [tex]\(51xy^2\)[/tex], the exponent of [tex]\(x\)[/tex] is [tex]\(1\)[/tex].
The GCF of the exponents of [tex]\(x\)[/tex] is the smallest exponent, which is [tex]\(1\)[/tex].
#### For [tex]\(y\)[/tex]:
- In [tex]\(27x^4y^3\)[/tex], the exponent of [tex]\(y\)[/tex] is [tex]\(3\)[/tex].
- In [tex]\(12x^2y\)[/tex], the exponent of [tex]\(y\)[/tex] is [tex]\(1\)[/tex].
- In [tex]\(51xy^2\)[/tex], the exponent of [tex]\(y\)[/tex] is [tex]\(2\)[/tex].
The GCF of the exponents of [tex]\(y\)[/tex] is the smallest exponent, which is [tex]\(1\)[/tex].
### Step 3: Combine the results
Combine the GCF of the coefficients and the GCF of the variable parts:
- Coefficient GCF: [tex]\(3\)[/tex]
- Variable part: [tex]\(x^1 y^1 = xy\)[/tex]
Thus, the GCF of the three monomials is:
[tex]\[ 3xy \][/tex]
The final answer is [tex]\( \boxed{3xy} \)[/tex].
