Find the square root of the perfect square made by multiplying 4050 with a least positive integer.

Options

A. 80
B. 90
C. 85
D. 95

Answer

To find the square root of the perfect square made by multiplying 4050 with the least positive integer, we will follow these steps:

### Step 1: Factorize 4050
First, we need to factorize 4050 into its prime factors.

- Divide by 2:
\[ 4050 \div 2 = 2025 \]
- Divide by 3:
\[ 2025 \div 3 = 675 \]
- Divide by 3 again:
\[ 675 \div 3 = 225 \]
- Divide by 3 again:
\[ 225 \div 3 = 75 \]
- Divide by 3 again:
\[ 75 \div 3 = 25 \]
- Divide by 5:
\[ 25 \div 5 = 5 \]
- Divide by 5 again:
\[ 5 \div 5 = 1 \]

So, the prime factorization of 4050 is:
\[ 4050 = 2^1 \times 3^4 \times 5^2 \]

### Step 2: Identify Missing Factors for Perfect Square
A perfect square has all prime factors raised to an even power. In our factorization:
- \(2^1\) (odd power)
- \(3^4\) (even power)
- \(5^2\) (even power)

The missing factor is \(2^1\). To make the power of 2 even, we need to multiply by \(2^1\).

### Step 3: Multiply by the Least Positive Integer
To make 4050 a perfect square, we need to multiply it by \(2^1\):
\[ \text{Least positive integer} = 2 \]
Thus, we multiply:
\[ 4050 \times 2 = 8100 \]

### Step 4: Find the Square Root of the Perfect Square
Now, we need to find the square root of 8100:
\[ \sqrt{8100} \]

Using the prime factorization:
\[ 8100 = 2^2 \times 3^4 \times 5^2 \]
Taking the square root:
\[ \sqrt{8100} = \sqrt{2^2} \times \sqrt{3^4} \times \sqrt{5^2} = 2 \times 3^2 \times 5 = 2 \times 9 \times 5 \]

Calculating this:
\[ 2 \times 9 = 18 \]
\[ 18 \times 5 = 90 \]

Thus, the square root of the perfect square is:
\[ \sqrt{8100} = 90 \]

### Final Answer
The square root of the perfect square made by multiplying 4050 with the least positive integer is **90**.

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