By which least possible number we divide to the 11760 so the resultant number becomes a perfect square :

Options

A. 3
B. 15
C. 7
D. Can't be determined

Answer

To determine the least possible number by which we can divide 11760 to make it a perfect square, we can follow these steps:

### Step 1: Prime Factorization of 11760
First, we need to find the prime factorization of 11760.

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

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

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

- Divide by 7:
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1 (stop here as we reached 1)

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

### Step 2: Identify the Exponents of the Prime Factors
Next, we look at the exponents in the prime factorization:
- For \(2^4\), the exponent is 4 (even).
- For \(3^1\), the exponent is 1 (odd).
- For \(5^1\), the exponent is 1 (odd).
- For \(7^2\), the exponent is 2 (even).

### Step 3: Make All Exponents Even
To make 11760 a perfect square, all the exponents in its prime factorization must be even.

- The exponent of 3 is 1 (odd), so we need to multiply by \(3^1\) to make it even.
- The exponent of 5 is 1 (odd), so we need to multiply by \(5^1\) to make it even.

### Step 4: Calculate the Least Number to Divide
Now, we need to find the product of the numbers we need to multiply to make the exponents even:
\[ 3^1 \times 5^1 = 3 \times 5 = 15 \]

### Step 5: Conclusion
Thus, the least possible number we need to divide 11760 by to make it a perfect square is **15**.

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