What least number must be multiplied to 3456 so that the priduct decomes a perfect cube?

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the problem of finding the least number that must be multiplied to 3456 so that the product becomes a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 3456
We need to find the prime factorization of 3456. We can do this by dividing the number by prime numbers.

1. Divide 3456 by 2:
- 3456 ÷ 2 = 1728
2. Divide 1728 by 2:
- 1728 ÷ 2 = 864
3. Divide 864 by 2:
- 864 ÷ 2 = 432
4. Divide 432 by 2:
- 432 ÷ 2 = 216
5. Divide 216 by 2:
- 216 ÷ 2 = 108
6. Divide 108 by 2:
- 108 ÷ 2 = 54
7. Divide 54 by 2:
- 54 ÷ 2 = 27
8. Divide 27 by 3:
- 27 ÷ 3 = 9
9. Divide 9 by 3:
- 9 ÷ 3 = 3
10. Divide 3 by 3:
- 3 ÷ 3 = 1

So, the prime factorization of 3456 is:
\[ 3456 = 2^6 \times 3^3 \]

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

- In our factorization \( 2^6 \times 3^3 \):
- The exponent of 2 is 6, which is already a multiple of 3.
- The exponent of 3 is 3, which is also a multiple of 3.

### Step 3: Determine the Missing Factors
Since both exponents are multiples of 3, we check if we need to adjust any of them to make them perfect cubes.

- The exponent of 2 (6) is already a multiple of 3.
- The exponent of 3 (3) is also a multiple of 3.

### Step 4: Conclusion
Since both prime factors already have exponents that are multiples of 3, we do not need to multiply by any additional numbers to make the product a perfect cube.

Thus, the least number that must be multiplied to 3456 to make it a perfect cube is:
\[ \text{Required number} = 1 \]

### Final Answer
The least number that must be multiplied to 3456 to make it a perfect cube is **1**.

---