By what least number should 4320 be multiplied so as to obtain a number which is a perfect cube ?

Options

A. a) 40
B. b) 50
C. c) 60
D. d) 80

Answer

To find the least number by which 4320 should be multiplied to obtain a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 4320
First, we need to find the prime factorization of 4320.

- Divide by 2:
\( 4320 \div 2 = 2160 \)
\( 2160 \div 2 = 1080 \)
\( 1080 \div 2 = 540 \)
\( 540 \div 2 = 270 \)
\( 270 \div 2 = 135 \)
- Now, divide by 3:
\( 135 \div 3 = 45 \)
\( 45 \div 3 = 15 \)
\( 15 \div 3 = 5 \)
- Finally, we have:
\( 5 \div 5 = 1 \)

So, the prime factorization of 4320 is:
\[ 4320 = 2^5 \times 3^3 \times 5^1 \]

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

- For \( 2^5 \): The nearest multiple of 3 is 6. We need \( 6 - 5 = 1 \) more \( 2 \).
- For \( 3^3 \): The exponent is already 3, which is a multiple of 3. No additional \( 3 \) is needed.
- For \( 5^1 \): The nearest multiple of 3 is 3. We need \( 3 - 1 = 2 \) more \( 5 \).

### Step 3: Calculate the Least Number to Multiply
Now, we need to multiply \( 4320 \) by the least number that will make all the exponents multiples of 3.

This means we need to multiply by:
\[ 2^1 \times 5^2 = 2 \times 25 = 50 \]

### Conclusion
Thus, the least number by which 4320 should be multiplied to obtain a perfect cube is **50**.

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