Common factor of `17 abc, 34 ab ^(2), 51 a ^(2)b` is

Options

A. `17 abc`
B. `17 ab `
C. `17 ac`
D. `17 a ^(2) b ^(2) c`

Answer

To find the common factor of the expressions \(17abc\), \(34ab^2\), and \(51a^2b\), we can follow these steps:

### Step 1: Factor each term
First, we will factor each of the given terms to identify their components.

1. **For \(17abc\)**:
\[ 17abc = 17 \times a \times b \times c \]

2. **For \(34ab^2\)**:
\[ 34ab^2 = 34 \times a \times b \times b = 34 \times a \times b^2 \]
Since \(34 = 2 \times 17\), we can rewrite it as:
\[ 34ab^2 = 2 \times 17 \times a \times b^2 \]

3. **For \(51a^2b\)**:
\[ 51a^2b = 51 \times a \times a \times b \]
Since \(51 = 3 \times 17\), we can rewrite it as:
\[ 51a^2b = 3 \times 17 \times a^2 \times b \]

### Step 2: Identify the common factors
Now, we will identify the common factors from the factored forms of each term.

- The numerical coefficients are \(17\), \(34\), and \(51\):
- The common factor among \(17\), \(2 \times 17\), and \(3 \times 17\) is \(17\).

- The variable \(a\):
- The first term has \(a^1\), the second term has \(a^1\), and the third term has \(a^2\). The common factor is \(a^1 = a\).

- The variable \(b\):
- The first term has \(b^1\), the second term has \(b^2\), and the third term has \(b^1\). The common factor is \(b^1 = b\).

- The variable \(c\):
- The first term has \(c^1\), while the other two terms do not have \(c\). Thus, \(c\) is not a common factor.

### Step 3: Combine the common factors
Now we can combine the common factors we identified:
\[ \text{Common Factor} = 17 \times a \times b = 17ab \]

### Final Answer
The common factor of \(17abc\), \(34ab^2\), and \(51a^2b\) is:
\[ \boxed{17ab} \]