What is the least positive integer by which 4500 should be divided so that the quotient is a perfect cube?

Options

A. 6
B. 36
C. 2
D. 3

Answer

To find the least positive integer by which 4500 should be divided so that the quotient is a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 4500
First, we need to find the prime factorization of 4500.

- 4500 can be divided by 10:
\[ 4500 \div 10 = 450 \]
- 450 can be divided by 10 again:
\[ 450 \div 10 = 45 \]
- 45 can be divided by 5:
\[ 45 \div 5 = 9 \]
- 9 can be divided by 3:
\[ 9 \div 3 = 3 \]
- Finally, 3 is a prime number.

So, we have:
\[ 4500 = 10 \times 10 \times 5 \times 3 \times 3 = 2^2 \times 3^2 \times 5^3 \]

### Step 2: Expressing the Factorization
Now we can express the prime factorization of 4500:
\[ 4500 = 2^2 \times 3^2 \times 5^3 \]

### Step 3: Conditions for a Perfect Cube
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 3 is 2 (not a multiple of 3).
- The exponent of 5 is 3 (is a multiple of 3).

### Step 4: Finding the Required Divisor
To make the exponents multiples of 3:
- For \(2^2\), we need to reduce it to \(2^0\) (which means we need to divide by \(2^2\)).
- For \(3^2\), we need to reduce it to \(3^0\) (which means we need to divide by \(3^2\)).
- The exponent of 5 is already fine.

Thus, we need to divide by:
\[ 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 5: Conclusion
The least positive integer by which 4500 should be divided so that the quotient is a perfect cube is **36**.