The least possible natural number by which if we multiply to the 1372, we get perfect cube number is :

Options

A. 2
B. 3
C. 5
D. can't be determined

Answer

To find the least possible natural number by which we can multiply 1372 to obtain a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 1372
First, we need to find the prime factorization of 1372.

1. Divide 1372 by 2 (the smallest prime number):
- 1372 ÷ 2 = 686
2. Divide 686 by 2:
- 686 ÷ 2 = 343
3. Now, 343 is not divisible by 2. Next, we try dividing by 7 (the next prime number):
- 343 ÷ 7 = 49
4. Divide 49 by 7:
- 49 ÷ 7 = 7
5. Finally, divide 7 by 7:
- 7 ÷ 7 = 1

So, the prime factorization of 1372 is:
\[ 1372 = 2^2 \times 7^3 \]

### Step 2: Analyze the Exponents
To form a perfect cube, all the exponents in the prime factorization must be multiples of 3.

- The exponent of 2 is 2 (not a multiple of 3).
- The exponent of 7 is 3 (a multiple of 3).

### Step 3: Determine the Required Multiplication
To make the exponent of 2 a multiple of 3, we need to increase it from 2 to 3. This means we need one more factor of 2.

Thus, we need to multiply 1372 by \( 2^{(3-2)} = 2^1 = 2 \).

### Step 4: Conclusion
The least possible natural number by which we need to multiply 1372 to get a perfect cube is:
\[ \text{Answer} = 2 \]