What is the least number which must be multiplied to 5400 to get a perfect square?

Options

A. 2
B. 3
C. 6
D. 10

Answer

To find the least number that must be multiplied to 5400 to get a perfect square, we will first find the prime factorization of 5400.

### Step 1: Prime Factorization of 5400
We start by dividing 5400 by the smallest prime number, which is 2.

- **Divide by 2**:
\( 5400 \div 2 = 2700 \)
- **Divide by 2 again**:
\( 2700 \div 2 = 1350 \)
- **Divide by 2 again**:
\( 1350 \div 2 = 675 \)

Now, 675 is not divisible by 2, so we move to the next prime number, which is 3.

- **Divide by 3**:
\( 675 \div 3 = 225 \)
- **Divide by 3 again**:
\( 225 \div 3 = 75 \)
- **Divide by 3 again**:
\( 75 \div 3 = 25 \)

Now, 25 is not divisible by 3, so we move to the next prime number, which is 5.

- **Divide by 5**:
\( 25 \div 5 = 5 \)
- **Divide by 5 again**:
\( 5 \div 5 = 1 \)

Now we have completed the factorization. The prime factorization of 5400 is:

\[ 5400 = 2^3 \times 3^3 \times 5^2 \]

### Step 2: Determine the Exponents
To form a perfect square, all the exponents in the prime factorization must be even.

- For \(2^3\), the exponent is 3 (odd).
- For \(3^3\), the exponent is 3 (odd).
- For \(5^2\), the exponent is 2 (even).

### Step 3: Adjust the Exponents
We need to make the exponents even:

- For \(2^3\): We need one more \(2\) to make it \(2^4\) (even).
- For \(3^3\): We need one more \(3\) to make it \(3^4\) (even).
- For \(5^2\): No additional \(5\) is needed since it is already even.

### Step 4: Calculate the Least Number to Multiply
To make the exponents even, we need to multiply by \(2^1\) and \(3^1\):

\[ \text{Least number} = 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Conclusion
The least number that must be multiplied to 5400 to get a perfect square is **6**.

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