The lengths of the sides of a traingle are `log_(10)12,log_(10)75andlog_(10)n`, where `n in N`. If a and b are the least ad greatest values of n respectively. The value of b-a is divisible by

Options

A. 221
B. 222
C. 223
D. 224

Answer

To solve the problem step by step, we need to determine the values of \( n \) such that the lengths of the sides of the triangle, given as \( \log_{10} 12 \), \( \log_{10} 75 \), and \( \log_{10} n \), satisfy the triangle inequality conditions.

### Step 1: Define the variables
Let:
- \( x = \log_{10} 12 \)
- \( y = \log_{10} 75 \)
- \( z = \log_{10} n \)

### Step 2: Apply the triangle inequality
We know that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we can write the following inequalities:

1. \( x + y > z \)
2. \( x + z > y \)
3. \( y + z > x \)

### Step 3: Solve the first inequality
From the first inequality \( x + y > z \):
\[ \log_{10} 12 + \log_{10} 75 > \log_{10} n \]
Using the property of logarithms \( \log_{10} a + \log_{10} b = \log_{10} (a \cdot b) \):
\[ \log_{10} (12 \cdot 75) > \log_{10} n \]
Calculating \( 12 \cdot 75 = 900 \):
\[ \log_{10} 900 > \log_{10} n \]
This implies:
\[ n < 900 \]

### Step 4: Solve the second inequality
From the second inequality \( x + z > y \):
\[ \log_{10} 12 + \log_{10} n > \log_{10} 75 \]
Using logarithmic properties:
\[ \log_{10} (12n) > \log_{10} 75 \]
This implies:
\[ 12n > 75 \quad \Rightarrow \quad n > \frac{75}{12} = 6.25 \]

### Step 5: Solve the third inequality
From the third inequality \( y + z > x \):
\[ \log_{10} 75 + \log_{10} n > \log_{10} 12 \]
Using logarithmic properties:
\[ \log_{10} (75n) > \log_{10} 12 \]
This implies:
\[ 75n > 12 \quad \Rightarrow \quad n > \frac{12}{75} = 0.16 \]

### Step 6: Combine the inequalities
From the inequalities we have:
1. \( n < 900 \)
2. \( n > 6.25 \)
3. \( n > 0.16 \)

The most restrictive conditions are:
\[ 6.25 < n < 900 \]

### Step 7: Determine the natural number values of \( n \)
Since \( n \) must be a natural number, the smallest integer greater than 6.25 is 7, and the largest integer less than 900 is 899. Thus:
- The least value \( a = 7 \)
- The greatest value \( b = 899 \)

### Step 8: Calculate \( b - a \)
Now, we calculate:
\[ b - a = 899 - 7 = 892 \]

### Step 9: Check divisibility
We need to find out if \( 892 \) is divisible by any of the options provided.

### Final Answer
The value of \( b - a \) is \( 892 \), and it is divisible by \( 23 \).