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 obtain a perfect square, we can follow these steps:

### Step 1: Prime Factorization of 5400
First, we need to perform the prime factorization of 5400.

1. Divide 5400 by 10:
\[ 5400 \div 10 = 540 \]
(10 = 2 × 5)

2. Divide 540 by 10:
\[ 540 \div 10 = 54 \]
(10 = 2 × 5)

3. Now, factor 54:
\[ 54 = 2 \times 27 = 2 \times 3^3 \]

So, we can write:
\[ 5400 = 10 \times 10 \times 54 = (2 \times 5) \times (2 \times 5) \times (2 \times 3^3) = 2^3 \times 3^3 \times 5^2 \]

### Step 2: Write the Prime Factorization
Now, we can write the complete prime factorization of 5400:
\[ 5400 = 2^3 \times 3^3 \times 5^2 \]

### Step 3: Identify the Exponents
Next, we look at the exponents of each prime factor:
- For \(2^3\), the exponent is 3 (which is odd).
- For \(3^3\), the exponent is 3 (which is odd).
- For \(5^2\), the exponent is 2 (which is even).

### Step 4: Make the Exponents Even
To make the number a perfect square, all the exponents in its prime factorization must be 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 \(5^2\), it is already even.

### Step 5: Calculate the Least Number
Now, we multiply the required factors:
\[ \text{Required number} = 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Conclusion
Thus, the least number which must be multiplied to 5400 to get a perfect square is **6**.

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