`a`, `b`,`c` are three complex numbers on the unit circle `|z|=1`, such that `abc=a+b+c`. Then `|ab+bc+ca|` is equal to

Options

A. `3`
B. `6`
C. `1`
D. `2`

Explanation

To solve the problem, we need to find the value of \(|ab + bc + ca|\) given that \(a\), \(b\), and \(c\) are complex numbers on the unit circle such that \(abc = a + b + c\).

### Step-by-Step Solution:

1. **Understanding the Unit Circle**:
Since \(a\), \(b\), and \(c\) are on the unit circle, we have:
\[ |a| = |b| = |c| = 1 \]

2. **Using the Given Condition**:
We know from the problem statement that:
\[ abc = a + b + c \]

3. **Finding the Modulus of \(ab + bc + ca\)**:
We want to find \(|ab + bc + ca|\). To do this, we can manipulate the expression using the properties of complex numbers.

4. **Multiplying by Conjugates**:
We can express \(ab\) in terms of its conjugate:
\[ |ab| = |a||b| = 1 \quad \text{(since both are on the unit circle)} \]
Therefore, we can write:
\[ |ab| = |a \cdot b| = |a| \cdot |b| = 1 \]
Similarly, \(|bc| = 1\) and \(|ca| = 1\).

5. **Using the Condition \(abc = a + b + c\)**:
We can rewrite the equation:
\[ 1 = \frac{a}{abc} + \frac{b}{abc} + \frac{c}{abc} \]
This implies:
\[ \frac{1}{bc} + \frac{1}{ca} + \frac{1}{ab} = 1 \]

6. **Finding the Modulus**:
Now, we can express:
\[ |ab + bc + ca| = |ab + bc + ca| = |1| \]
Since we have established that the sum of the reciprocals equals 1, we can conclude:
\[ |ab + bc + ca| = 1 \]

### Final Answer:
Thus, the value of \(|ab + bc + ca|\) is:
\[ \boxed{1} \]