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 performing the prime factorization of 5808.

1. Divide by 2:
- 5808 ÷ 2 = 2904
2. Divide by 2 again:
- 2904 ÷ 2 = 1452
3. Divide by 2 again:
- 1452 ÷ 2 = 726
4. Divide by 2 again:
- 726 ÷ 2 = 363
5. Now, divide by 3:
- 363 ÷ 3 = 121
6. Finally, divide by 11:
- 121 ÷ 11 = 11
7. Divide by 11 again:
- 11 ÷ 11 = 1

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

### Step 2: Identify the Exponents
Next, we look at the exponents of the prime factors:
- For \(2^4\), the exponent is 4 (even).
- For \(3^1\), the exponent is 1 (odd).
- For \(11^2\), the exponent is 2 (even).

### Step 3: Make Exponents Even
To make 5808 a perfect square, all the exponents in its prime factorization must be even.

- The exponent of 2 is already even (4).
- The exponent of 3 is odd (1), so we need one more 3 to make it even (1 + 1 = 2).
- The exponent of 11 is already even (2).

### Step 4: Determine the Smallest Number to Multiply
Since we need one more 3 to make the exponent of 3 even, we multiply 5808 by 3.

### Step 5: Conclusion
Thus, the smallest number by which 5808 should be multiplied to become a perfect square is:
\[ \boxed{3} \]

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