If a is an irrational number which is divisible by b, then the number b

Options

A. must be rational
B. must be irrational
C. may be rational or irrational
D. None of these

Answer

To solve the problem, we need to analyze the relationship between the irrational number \( a \) and the number \( b \) which divides \( a \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given that \( a \) is an irrational number and \( b \) is a number that divides \( a \). We need to determine the nature of \( b \) (whether it can be rational, irrational, or both).

2. **Let’s Define an Example**:
Let's consider an irrational number. A common example is \( a = \sqrt{3} \).

3. **Divisibility by Rational Number**:
Now, let's take \( b = 2 \), which is a rational number. We can check if \( b \) divides \( a \):
\[ \frac{a}{b} = \frac{\sqrt{3}}{2} \]
This result \( \frac{\sqrt{3}}{2} \) is still an irrational number. Hence, \( b = 2 \) divides \( a \) and is rational.

4. **Divisibility by Irrational Number**:
Now, let’s consider \( b = \sqrt{3} \) itself. We can check if it divides \( a \):
\[ \frac{a}{b} = \frac{\sqrt{3}}{\sqrt{3}} = 1 \]
Here, \( 1 \) is a rational number. So, \( b = \sqrt{3} \) divides \( a \) and is irrational.

5. **Conclusion**:
From the examples above, we see that \( b \) can be either a rational number (like \( 2 \)) or an irrational number (like \( \sqrt{3} \)). Therefore, we conclude that \( b \) can be either rational or irrational.

### Final Answer:
The number \( b \) can be either rational or irrational.