The least number which when divides 37044, gives the result a perfect cube, is :

Options

A. 2
B. 4
C. 14
D. 21

Answer

To find the least number which, when divides 37044, results in a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 37044
First, we need to find the prime factorization of 37044.

- Divide by 2:
\[ 37044 \div 2 = 18522 \]
- Divide by 2 again:
\[ 18522 \div 2 = 9261 \]
- Divide by 3:
\[ 9261 \div 3 = 3087 \]
- Divide by 3 again:
\[ 3087 \div 3 = 1029 \]
- Divide by 3 again:
\[ 1029 \div 3 = 343 \]
- Divide by 7:
\[ 343 \div 7 = 49 \]
- Divide by 7 again:
\[ 49 \div 7 = 7 \]
- Divide by 7 again:
\[ 7 \div 7 = 1 \]

Thus, the prime factorization of 37044 is:
\[ 37044 = 2^2 \times 3^3 \times 7^3 \]

### Step 2: Identify the Exponents
Now we need to look at the exponents of the prime factors:
- For \(2\), the exponent is \(2\).
- For \(3\), the exponent is \(3\).
- For \(7\), the exponent is \(3\).

### Step 3: Determine the Required Exponents for a Perfect Cube
For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of \(3\):
- The exponent of \(2\) needs to be increased from \(2\) to \(3\) (needs \(1\) more).
- The exponent of \(3\) is already \(3\) (no change needed).
- The exponent of \(7\) is already \(3\) (no change needed).

### Step 4: Calculate the Least Number to Multiply
To make \(37044\) a perfect cube, we need to multiply it by \(2^1\) (which is \(2\)) to make the exponent of \(2\) equal to \(3\).

### Step 5: Conclusion
Thus, the least number which when divides \(37044\) gives a perfect cube is:
\[ \text{Least number} = 2 \]