What is the least number by which 30375 should be multiplied to get a perfect cube?

Answer

To find the least number by which 30375 should be multiplied to obtain a perfect cube, we can follow these steps:

### Step 1: Prime Factorization
First, we need to perform the prime factorization of the number 30375.

1. **Divide by 3**:
- 30375 ÷ 3 = 10125
- 10125 ÷ 3 = 3375
- 3375 ÷ 3 = 1125
- 1125 ÷ 3 = 375
- 375 ÷ 3 = 125
- 125 is not divisible by 3, so we move to the next prime number (5).

2. **Divide by 5**:
- 125 ÷ 5 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1

So, the prime factorization of 30375 is:
\[ 30375 = 3^5 \times 5^3 \]

### Step 2: Grouping the Factors
Next, we need to group the factors into sets of three (since we are looking for a perfect cube).

- The exponent of 3 in \(3^5\) is 5, which can be grouped as:
- \(3^3\) (which is a perfect cube) and leaves \(3^2\) (which is not a perfect cube).

- The exponent of 5 in \(5^3\) is 3, which is already a perfect cube.

### Step 3: Determine the Missing Factors
To make the entire expression a perfect cube, we need to ensure that all prime factors have exponents that are multiples of 3.

- For \(3^5\), we have \(5\) which is not a multiple of \(3\). We need \(3\) more \(3s\) to make it \(3^6\) (which is a perfect cube).
- For \(5^3\), it is already a perfect cube.

### Step 4: Calculate the Least Number to Multiply
Since we need one more \(3\) to make \(3^6\), we multiply by \(3\).

Thus, the least number by which 30375 should be multiplied to get a perfect cube is:
\[ \text{Least number} = 3 \]

### Final Answer
The least number by which 30375 should be multiplied to get a perfect cube is **3**.

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