The least number by which we multiply to the 11760, so that can get a perfect square number:

Options

A. 2
B. 3
C. 5
D. None of these

Answer

To find the least number by which we need to multiply 11760 to make it a perfect square, we can follow these steps:

### Step 1: Prime Factorization of 11760
We start by finding the prime factorization of 11760.

1. Divide by 2:
- 11760 ÷ 2 = 5880
- 5880 ÷ 2 = 2940
- 2940 ÷ 2 = 1470
- 1470 ÷ 2 = 735 (stop here as 735 is odd)

2. Divide by 3:
- 735 ÷ 3 = 245 (stop here as 245 is not divisible by 3)

3. Divide by 5:
- 245 ÷ 5 = 49 (stop here as 49 is not divisible by 5)

4. Divide by 7:
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1

So, the prime factorization of 11760 is:
\[ 11760 = 2^4 \times 3^1 \times 5^1 \times 7^2 \]

### Step 2: Identify the Exponents
Next, we look at the exponents of the prime factors:
- \(2^4\) (even)
- \(3^1\) (odd)
- \(5^1\) (odd)
- \(7^2\) (even)

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

- For \(3^1\), we need one more \(3\) to make it \(3^2\) (add 1).
- For \(5^1\), we need one more \(5\) to make it \(5^2\) (add 1).

### Step 4: Calculate the Least Number to Multiply
Thus, we need to multiply by:
\[ 3^1 \times 5^1 = 3 \times 5 = 15 \]

### Conclusion
The least number by which we need to multiply 11760 to make it a perfect square is **15**.

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