The smallest natural number, by which 3000 must be divided to make the quotient a perfect cube, is :

Options

A. 3
B. 4
C. 5
D. 6

Answer

To find the smallest natural number by which 3000 must be divided to make the quotient a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 3000
First, we need to find the prime factorization of 3000.

3000 can be factored as follows:
- 3000 = 3 × 1000
- 1000 = 10 × 100 = 10 × 10 × 10 = (2 × 5)³
- Therefore, 1000 = 2³ × 5³

Combining these, we have:
3000 = 3¹ × 2³ × 5³

### Step 2: Understand the Condition for a Perfect Cube
A number is a perfect cube if all the exponents in its prime factorization are multiples of 3.

### Step 3: Analyze the Exponents
From our factorization:
- The exponent of 3 is 1 (which is not a multiple of 3)
- The exponent of 2 is 3 (which is a multiple of 3)
- The exponent of 5 is 3 (which is a multiple of 3)

### Step 4: Determine the Smallest Natural Number to Divide
To make the exponent of 3 a multiple of 3, we need to reduce it from 1 to 0. This means we need to divide by 3.

### Step 5: Conclusion
Thus, the smallest natural number by which 3000 must be divided to make the quotient a perfect cube is **3**.

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