In a vernier calipers, `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 regarding the least count of a vernier caliper, we will follow these steps:

### Step 1: Understand the relationship between main scale and vernier scale
In a vernier caliper, `N` divisions of the main scale coincide with `N + m` divisions of the vernier scale. This gives us a relationship that we can use to express the value of one vernier scale division (VSD) in terms of main scale divisions (MSD).

### Step 2: Derive the formula for one vernier scale division
From the problem statement, we can express the value of one vernier scale division as:
\[ \text{VSD} = \frac{N}{N + m} \text{ (in terms of main scale division)} \]

### Step 3: Write the formula for least count
The least count (LC) of a vernier caliper is defined as:
\[ \text{LC} = \text{1 MSD} - \text{1 VSD} \]
Substituting the expression for VSD:
\[ \text{LC} = 1 - \frac{N}{N + m} \]

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

### Step 5: Analyze the least count for minimum value
To minimize the least count, we need to maximize the denominator (N + m). This means we should minimize the value of `m`. However, `m` cannot be zero because that would lead to a least count of zero, which is not acceptable.

### Step 6: Determine the minimum acceptable value of `m`
The smallest integer value for `m` that is greater than zero is `1`. Thus, we set:
\[ m = 1 \]

### Step 7: Conclusion
Therefore, the value of `m` for which the instrument has the minimum least count is:
\[ \boxed{1} \]

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