Let `log_(a)N=alpha + beta` where ` alpha ` is integer and ` beta =[0,1)`. Then , On the basis of above information , answer the following questions.
The difference of largest and smallest integral value of N satisfying ` alpha =3 and a =5` , is

Options

A. 499
B. 500
C. 501
D. 502

Answer

To solve the problem, we need to find the difference between the largest and smallest integral values of \( N \) that satisfy the given conditions. Let's break down the steps:

### Step 1: Understand the given equation
We have:
\[ \log_a N = \alpha + \beta \]
where \( \alpha \) is an integer and \( \beta \) is in the range \([0, 1)\).

### Step 2: Substitute the known values
We are given:
- \( \alpha = 3 \)
- \( a = 5 \)

Substituting these values into the equation, we get:
\[ \log_5 N = 3 + \beta \]

### Step 3: Convert the logarithmic equation to exponential form
Using the properties of logarithms, we can rewrite the equation as:
\[ N = 5^{3 + \beta} \]

### Step 4: Determine the range of \( N \)
Since \( \beta \) is in the range \([0, 1)\), we can find the range of \( N \):
- When \( \beta = 0 \):
\[ N = 5^{3 + 0} = 5^3 = 125 \]
- When \( \beta \) approaches \( 1 \) (but does not include it):
\[ N = 5^{3 + 1} = 5^4 = 625 \]

Since \( \beta \) cannot actually be \( 1 \), \( N \) can approach \( 625 \) but will never actually reach it. Therefore, the largest integral value of \( N \) is \( 624 \).

### Step 5: Identify the smallest and largest integral values
- The smallest integral value of \( N \) is \( 125 \).
- The largest integral value of \( N \) is \( 624 \).

### Step 6: Calculate the difference
Now, we find the difference between the largest and smallest integral values of \( N \):
\[ \text{Difference} = 624 - 125 = 499 \]

### Final Answer
The difference of the largest and smallest integral values of \( N \) satisfying the conditions is:
\[ \boxed{499} \]