By which smallest number should 5808 be multiplied so that it becomes a perfect square?

Options

A. 2
B. 7
C. 11
D. 3

Answer

To determine the smallest number by which 5808 should be multiplied to become a perfect square, we will follow these steps:

### Step 1: Prime Factorization of 5808
We start by finding the prime factorization of 5808.

- Divide 5808 by 2:
- 5808 ÷ 2 = 2904
- 2904 ÷ 2 = 1452
- 1452 ÷ 2 = 726
- 726 ÷ 2 = 363
- Now, 363 is not divisible by 2, so we try the next prime number, which is 3:
- 363 ÷ 3 = 121
- Next, we factor 121:
- 121 = 11 × 11 (or 11²)

Thus, the prime factorization of 5808 is:
\[ 5808 = 2^4 \times 3^1 \times 11^2 \]

### Step 2: Identify the Exponents
In the prime factorization, we have:
- \(2^4\) (even exponent)
- \(3^1\) (odd exponent)
- \(11^2\) (even exponent)

### Step 3: Determine the Odd Exponents
To form a perfect square, all the exponents in the prime factorization must be even. Here, the exponent of 3 is odd (1).

### Step 4: Find the Smallest Number to Multiply
To make the exponent of 3 even, we need to multiply by 3 (which will change the exponent from 1 to 2).

### Conclusion
Therefore, the smallest number by which 5808 should be multiplied to become a perfect square is:
\[ \text{Answer} = 3 \]