Find the least number by which 22932 must be multiplied or divided so as to make it a perfect square :

Options

A. a)10
B. b)4
C. c)11
D. d)13

Answer

To find the least number by which 22,932 must be multiplied or divided to make it a perfect square, we can follow these steps:

### Step 1: Prime Factorization
First, we need to perform the prime factorization of 22,932.

1. Divide by 2 (the smallest prime number):
- 22,932 ÷ 2 = 11,466
- 11,466 ÷ 2 = 5,733 (2 appears twice)

2. Next, divide by 3:
- 5,733 ÷ 3 = 1,911
- 1,911 ÷ 3 = 637 (3 appears twice)

3. Now, divide by 7:
- 637 ÷ 7 = 91 (7 appears once)

4. Finally, divide by 13:
- 91 ÷ 7 = 13 (7 appears twice)

So, the prime factorization of 22,932 is:
\[ 22,932 = 2^2 \times 3^2 \times 7^2 \times 13^1 \]

### Step 2: Identify the Exponents
In a perfect square, all prime factors must have even exponents. From the factorization:
- \(2^2\) (even)
- \(3^2\) (even)
- \(7^2\) (even)
- \(13^1\) (odd)

### Step 3: Make Exponents Even
To make the exponent of 13 even, we need to either multiply or divide by 13. Since the exponent of 13 is currently 1 (odd), we need one more 13 to make it 2 (even).

### Step 4: Conclusion
Thus, the least number by which 22,932 must be multiplied to make it a perfect square is 13.

### Final Answer
The least number is **13**.

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