There exists a natural number `N` which is 50 times its own logarithm to the base 10, then `N` is divisible by 5 (b) 7 (c) 9 (d) 11

Options

A. 5
B. 7
C. 9
D. 11

Explanation

To solve the problem, we need to find the natural number \( N \) such that \( N = 50 \log_{10} N \). We will follow these steps:

### Step 1: Set up the equation
Given the condition, we can write:
\[ N = 50 \log_{10} N \]

### Step 2: Rewrite the logarithmic equation
We can rearrange the equation to isolate the logarithm:
\[ \frac{N}{50} = \log_{10} N \]

### Step 3: Exponentiate both sides
Using the property of logarithms, we can exponentiate both sides to eliminate the logarithm:
\[ N = 10^{\frac{N}{50}} \]

### Step 4: Take the natural logarithm of both sides
Taking the natural logarithm (ln) of both sides gives us:
\[ \ln N = \frac{N}{50} \ln 10 \]

### Step 5: Rearrange the equation
Rearranging the equation, we have:
\[ \frac{\ln N}{N} = \frac{\ln 10}{50} \]

### Step 6: Analyze the function
Let \( f(x) = \frac{\ln x}{x} \). We need to find where this function equals \( \frac{\ln 10}{50} \).

### Step 7: Find the value of \( N \)
By trial or using numerical methods, we can find that \( N \) is approximately 100.

### Step 8: Check divisibility
Now we check which of the given options \( N = 100 \) is divisible by:
- \( 100 \div 5 = 20 \) (divisible)
- \( 100 \div 7 \approx 14.29 \) (not divisible)
- \( 100 \div 9 \approx 11.11 \) (not divisible)
- \( 100 \div 11 \approx 9.09 \) (not divisible)

Thus, \( N \) is divisible by 5.

### Conclusion
The answer is that \( N \) is divisible by **5**.

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