The least possible number by which if we divide 1372, it will become a perfect cube number is:

Options

A. 2
B. 7
C. 3
D. 4

Answer

To find the least possible number by which we can divide 1372 to make it a perfect cube, we need to follow these steps:

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

- Divide 1372 by 2 (the smallest prime number):
\[ 1372 \div 2 = 686 \]
- Divide 686 by 2:
\[ 686 \div 2 = 343 \]
- Now, 343 is not divisible by 2. Next, we try dividing by 7:
\[ 343 \div 7 = 49 \]
- Now, divide 49 by 7:
\[ 49 \div 7 = 7 \]
- Finally, divide 7 by 7:
\[ 7 \div 7 = 1 \]

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

### Step 2: Identify the Exponents
In order for a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.

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

### Step 3: Make the Exponent of 2 a Multiple of 3
To make the exponent of 2 a multiple of 3, we need to increase it from 2 to the next multiple of 3, which is 3.

This means we need one more factor of 2:
\[ 2^2 \rightarrow 2^3 \]

### Step 4: Determine the Least Number to Divide
Since we need to add one more factor of 2, we need to divide 1372 by 2:
\[ \text{Least number} = 2 \]

### Conclusion
The least possible number by which we can divide 1372 to make it a perfect cube is **2**.

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