Let `log_(3)N=alpha_(1)+beta_(1)`
`log_(5)N=alpha_(2)+beta_(2)`
`log_(7)N=alpha_(3)+beta_(3)`
where `alpha_(1), alpha_(2) and alpha_(3)` are integers and `beta_(1), beta_(2), beta_(3) in [0,1)`.
Q. Difference of largest and smallest values of N if `alpha_(1)=5, alpha_(2)=3 and alpha_(3)=2`.

Options

A. 97
B. 100
C. 98
D. 99

Answer

To solve the problem, we need to find the largest and smallest values of \( N \) based on the given logarithmic equations. Let's break down the solution step by step.

### Step 1: Understand the logarithmic equations
We have the following equations:
1. \( \log_3 N = \alpha_1 + \beta_1 \)
2. \( \log_5 N = \alpha_2 + \beta_2 \)
3. \( \log_7 N = \alpha_3 + \beta_3 \)

Given:
- \( \alpha_1 = 5 \)
- \( \alpha_2 = 3 \)
- \( \alpha_3 = 2 \)
- \( \beta_1, \beta_2, \beta_3 \in [0, 1) \)

### Step 2: Express \( N \) in terms of \( \alpha \) and \( \beta \)
From the logarithmic identities, we can express \( N \) as:
- From the first equation:
\[ N = 3^{\alpha_1 + \beta_1} = 3^5 \cdot 3^{\beta_1} \]
- From the second equation:
\[ N = 5^{\alpha_2 + \beta_2} = 5^3 \cdot 5^{\beta_2} \]
- From the third equation:
\[ N = 7^{\alpha_3 + \beta_3} = 7^2 \cdot 7^{\beta_3} \]

### Step 3: Calculate the bounds for \( N \)
1. **For \( N \) from the first equation:**
- Minimum value when \( \beta_1 = 0 \):
\[ N_{\text{min}} = 3^5 = 243 \]
- Maximum value when \( \beta_1 \) approaches 1:
\[ N_{\text{max}} < 3^{5 + 1} = 3^6 = 729 \]

2. **For \( N \) from the second equation:**
- Minimum value when \( \beta_2 = 0 \):
\[ N_{\text{min}} = 5^3 = 125 \]
- Maximum value when \( \beta_2 \) approaches 1:
\[ N_{\text{max}} < 5^{3 + 1} = 5^4 = 625 \]

3. **For \( N \) from the third equation:**
- Minimum value when \( \beta_3 = 0 \):
\[ N_{\text{min}} = 7^2 = 49 \]
- Maximum value when \( \beta_3 \) approaches 1:
\[ N_{\text{max}} < 7^{2 + 1} = 7^3 = 343 \]

### Step 4: Determine the overall bounds for \( N \)
- The smallest value of \( N \) is the maximum of the minimum values:
\[ \text{Smallest } N = \max(243, 125, 49) = 243 \]

- The largest value of \( N \) is the minimum of the maximum values:
\[ \text{Largest } N < \min(729, 625, 343) = 343 \]
However, since \( N \) cannot equal 343, we take the largest value as:
\[ \text{Largest } N = 342 \]

### Step 5: Calculate the difference
Now, we find the difference between the largest and smallest values of \( N \):
\[ \text{Difference} = 342 - 243 = 99 \]

### Final Answer
The difference of the largest and smallest values of \( N \) is **99**.