The least number, by which 1944 must be multiplied so as to make the result a perfect cube, is

Options

A. A) 2
B. B) 3
C. C) 6
D. D) 13

Answer

To find the least number by which 1944 must be multiplied to make it a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 1944
First, we need to factor 1944 into its prime factors.

- Divide 1944 by 2:
\[ 1944 \div 2 = 972 \]
- Divide 972 by 2:
\[ 972 \div 2 = 486 \]
- Divide 486 by 2:
\[ 486 \div 2 = 243 \]
- Now, 243 is not divisible by 2, so we divide by 3:
\[ 243 \div 3 = 81 \]
- Divide 81 by 3:
\[ 81 \div 3 = 27 \]
- Divide 27 by 3:
\[ 27 \div 3 = 9 \]
- Divide 9 by 3:
\[ 9 \div 3 = 3 \]
- Finally, divide 3 by 3:
\[ 3 \div 3 = 1 \]

So, the prime factorization of 1944 is:
\[ 1944 = 2^3 \times 3^5 \]

### Step 2: Identify the Exponents
In the prime factorization \(2^3 \times 3^5\), we have:
- The exponent of 2 is 3.
- The exponent of 3 is 5.

### Step 3: Adjust the Exponents to Make Them Multiples of 3
For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.

- The exponent of 2 (which is 3) is already a multiple of 3.
- The exponent of 3 (which is 5) needs to be adjusted. The nearest multiple of 3 greater than 5 is 6.

### Step 4: Determine the Factor Needed
To change the exponent of 3 from 5 to 6, we need to multiply by \(3^1\) (which is 3).

### Step 5: Conclusion
Thus, the least number by which 1944 must be multiplied to make it a perfect cube is:
\[ \text{Answer} = 3 \]

### Final Answer
The least number is **3**.

---