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 can follow these steps:

### Step 1: Factor the Number
First, we need to find the prime factorization of 5808.

1. Start dividing by the smallest prime number, which is 2:
- 5808 ÷ 2 = 2904
- 2904 ÷ 2 = 1452
- 1452 ÷ 2 = 726
- 726 ÷ 2 = 363
- 363 is not divisible by 2, so we move to the next prime number, which is 3:
- 363 ÷ 3 = 121
- 121 is not divisible by 3, so we move to the next prime number, which is 11:
- 121 ÷ 11 = 11
- 11 ÷ 11 = 1

Thus, 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 in the prime factorization:
- For \(2\), the exponent is 4 (which is even).
- For \(3\), the exponent is 1 (which is odd).
- For \(11\), the exponent is 2 (which is even).

### Step 3: Make Exponents Even
A perfect square requires all prime factors to have even exponents. Here, the exponent of \(3\) is odd. To make it even, we need one more \(3\).

### Step 4: Calculate the Required Multiplier
To make the exponent of \(3\) even, we need to multiply by \(3\):
\[ \text{Required multiplier} = 3 \]

### Conclusion
Thus, the smallest number by which 5808 should be multiplied to become a perfect square is \(3\).

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