By what least number `25 xx 20 xx 9 xx 12 xx 30` should be multiplied to make it a perfect square number?

Options

A. 5
B. 4
C. 3
D. 2

Answer

To determine the least number by which \( 25 \times 20 \times 9 \times 12 \times 30 \) should be multiplied to make it a perfect square, we will follow these steps:

### Step 1: Prime Factorization
First, we will find the prime factorization of each number in the product.

- \( 25 = 5^2 \)
- \( 20 = 2^2 \times 5^1 \)
- \( 9 = 3^2 \)
- \( 12 = 2^2 \times 3^1 \)
- \( 30 = 2^1 \times 3^1 \times 5^1 \)

### Step 2: Combine the Prime Factors
Now, we will combine all the prime factors together:

\[ 25 \times 20 \times 9 \times 12 \times 30 = 5^2 \times (2^2 \times 5^1) \times 3^2 \times (2^2 \times 3^1) \times (2^1 \times 3^1 \times 5^1) \]

Combining the factors:

- For \( 2 \): \( 2^2 + 2^2 + 1 = 5 \) (total exponent of 2)
- For \( 3 \): \( 0 + 0 + 2 + 1 + 1 = 4 \) (total exponent of 3)
- For \( 5 \): \( 2 + 1 + 0 + 0 + 1 = 4 \) (total exponent of 5)

Thus, we have:

\[ 25 \times 20 \times 9 \times 12 \times 30 = 2^5 \times 3^4 \times 5^4 \]

### Step 3: Check for Perfect Square
A number is a perfect square if all the exponents in its prime factorization are even.

- The exponent of \( 2 \) is \( 5 \) (odd)
- The exponent of \( 3 \) is \( 4 \) (even)
- The exponent of \( 5 \) is \( 4 \) (even)

### Step 4: Make the Exponents Even
To make the exponent of \( 2 \) even, we need to multiply by \( 2^1 \) (to increase it from \( 5 \) to \( 6 \)).

### Conclusion
Thus, the least number we need to multiply \( 25 \times 20 \times 9 \times 12 \times 30 \) by to make it a perfect square is \( 2 \).

### Final Answer
The least number is \( 2 \).

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