In a vernier callipers, `N` divisions of the main scale coincide with `N + m` divisions of the vernier scale. what is the value of `m` for which the instrument has minimum least count.

Explanation

To solve the problem, we need to determine the value of \( m \) for which the least count of the vernier calipers is minimized. Here’s the step-by-step solution:

### Step 1: Understand the relationship between the main scale and the vernier scale
We are given that \( N \) divisions of the main scale coincide with \( N + m \) divisions of the vernier scale. This can be expressed as:
\[ N \text{ MS divisions} = (N + m) \text{ VS divisions} \]

### Step 2: Determine the value of one vernier scale division
From the above relationship, we can find the value of one vernier scale division (VSD):
\[ 1 \text{ VSD} = \frac{N}{N + m} \text{ MS divisions} \]

### Step 3: Define the least count of the vernier calipers
The least count (LC) of the vernier calipers is defined as:
\[ \text{LC} = 1 \text{ MS division} - 1 \text{ VSD} \]
Substituting the value of VSD from Step 2, we get:
\[ \text{LC} = 1 - \frac{N}{N + m} \]

### Step 4: Simplify the least count expression
Now, we simplify the expression for the least count:
\[ \text{LC} = 1 - \frac{N}{N + m} = \frac{(N + m) - N}{N + m} = \frac{m}{N + m} \]

### Step 5: Minimize the least count
To minimize the least count, we need to maximize the fraction \( \frac{m}{N + m} \). Since \( N \) is constant, we can analyze the behavior of this fraction as \( m \) changes.

### Step 6: Analyze the function
The fraction \( \frac{m}{N + m} \) increases as \( m \) increases. However, if \( m = 0 \), the least count becomes zero, which is not acceptable. Therefore, we need to find the smallest positive integer value of \( m \) that gives a finite least count.

### Step 7: Find the optimal value of \( m \)
The smallest positive integer value for \( m \) is \( 1 \). Thus, substituting \( m = 1 \) into our least count expression:
\[ \text{LC} = \frac{1}{N + 1} \]
This gives us the minimum least count for the vernier calipers.

### Final Answer
The value of \( m \) for which the instrument has the minimum least count is:
\[ \boxed{1} \]