What is the minimum number you should multiply 648 with to get a perfect square?

Options

A. 2
B. 3
C. 5
D. 7

Answer

To find the minimum number that should be multiplied with 648 to make it a perfect square, we will follow these steps:

### Step 1: Prime Factorization of 648
First, we need to perform the prime factorization of 648.

648 can be divided by 2:
- 648 ÷ 2 = 324
- 324 ÷ 2 = 162
- 162 ÷ 2 = 81
- 81 is not divisible by 2, so we switch to the next prime number, which is 3.
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 648 is:
\[ 648 = 2^3 \times 3^4 \]

### Step 2: Identify the Exponents
Next, we look at the exponents in the prime factorization:
- For \(2^3\), the exponent is 3 (which is odd).
- For \(3^4\), the exponent is 4 (which is even).

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

- The exponent of 2 is 3 (odd), so we need to multiply by \(2^1\) to make it \(2^4\) (even).
- The exponent of 3 is already 4 (even), so we do not need to multiply by anything for 3.

### Step 4: Calculate the Minimum Number
Thus, the minimum number we need to multiply 648 by to make it a perfect square is:
\[ 2^1 = 2 \]

### Conclusion
Therefore, the minimum number you should multiply 648 with to get a perfect square is **2**.

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