संख्या 42592 मैं किस छोटी से छोटी संख्या से भाग किया जाए ताकि भागफल एक पूर्ण वर्ग बन जाए -

Options

A. 17
B. 13
C. 22
D. 15

Answer

To solve the question, "संख्या 42592 मैं किस छोटी से छोटी संख्या से भाग किया जाए ताकि भागफल एक पूर्ण वर्ग बन जाए", we will follow these steps:

### Step 1: Understand the Problem
We need to find the smallest number that can be divided into 42592 such that the result is a perfect square.

### Step 2: Factorize the Number
First, we will perform the prime factorization of 42592.

1. **Divide by 2 (since 42592 is even):**
- 42592 ÷ 2 = 21296
- 21296 ÷ 2 = 10648
- 10648 ÷ 2 = 5324
- 5324 ÷ 2 = 2662
- 2662 ÷ 2 = 1331

2. **Now, 1331 is not divisible by 2. Let's check for 3:**
- The sum of digits (1 + 3 + 3 + 1 = 8) is not divisible by 3, so we move to the next prime number.

3. **Check for 7:**
- 1331 ÷ 7 = 190.14 (not divisible)

4. **Check for 11:**
- 1331 ÷ 11 = 121 (which is 11 x 11)

So, the prime factorization of 42592 is:
\[ 42592 = 2^5 \times 11^3 \]

### Step 3: Identify the Perfect Square Condition
For a number to be a perfect square, all the powers in its prime factorization must be even.

- In our factorization:
- The power of 2 is 5 (which is odd)
- The power of 11 is 3 (which is also odd)

### Step 4: Make the Powers Even
To make the powers even:
- For \(2^5\), we need one more 2 to make it \(2^6\).
- For \(11^3\), we need one more 11 to make it \(11^4\).

### Step 5: Calculate the Smallest Number to Divide
To achieve this, we need to multiply the number by \(2^1 \times 11^1\):
\[ 2^1 \times 11^1 = 2 \times 11 = 22 \]

### Step 6: Conclusion
Now, we can divide 42592 by 22 to check if the result is a perfect square:
\[ \frac{42592}{22} = 1936 \]
Since \(1936 = 44^2\), it is indeed a perfect square.

Thus, the smallest number we need to divide by is:
**22**