Let a and b be natural numbers, not necessarily distinct. For all values of a and b, the natural number would be

Options

A. (a +b)
B. `a/b`
C. (a - b)
D. log (ab)

Answer

To solve the problem, we need to analyze the four options given for the natural numbers \( a \) and \( b \). Let's go through each option step by step.

### Step-by-Step Solution:

1. **Understanding Natural Numbers**:
- Natural numbers are the set of positive integers starting from 1, i.e., \( \{1, 2, 3, \ldots\} \).

2. **Option 1: \( a + b \)**:
- When we add two natural numbers \( a \) and \( b \), the result is always a natural number.
- Example: If \( a = 2 \) and \( b = 3 \), then \( a + b = 2 + 3 = 5 \), which is natural.
- Conclusion: This option is **correct**.

3. **Option 2: \( \frac{a}{b} \)**:
- The division of two natural numbers does not always yield a natural number.
- Example: If \( a = 2 \) and \( b = 3 \), then \( \frac{a}{b} = \frac{2}{3} \), which is not a natural number.
- Conclusion: This option is **incorrect**.

4. **Option 3: \( a - b \)**:
- The subtraction of two natural numbers can result in a negative number or zero, which are not natural numbers.
- Example: If \( a = 2 \) and \( b = 3 \), then \( a - b = 2 - 3 = -1 \), which is not natural.
- Conclusion: This option is **incorrect**.

5. **Option 4: \( \log(a \times b) \)**:
- The product of two natural numbers \( a \) and \( b \) is a natural number, but the logarithm of a natural number is not necessarily a natural number.
- Example: If \( a = 2 \) and \( b = 3 \), then \( \log(2 \times 3) = \log(6) \), which is not a natural number.
- Conclusion: This option is **incorrect**.

### Final Conclusion:
The only option that is guaranteed to be a natural number for all values of \( a \) and \( b \) is **Option 1: \( a + b \)**.