Find the smallest number by which 210125 must be multiped so that the product is a perfect cube.

Answer

To find the smallest number by which 210125 must be multiplied so that the product is a perfect cube, we will follow these steps:

### Step 1: Factorize 210125
First, we need to find the prime factorization of 210125.

- The last digit is 5, so we can divide by 5:
- 210125 ÷ 5 = 42025
- 42025 ÷ 5 = 8405
- 8405 ÷ 5 = 1681

Now we have:
\[ 210125 = 5^3 \times 1681 \]

### Step 2: Factorize 1681
Next, we need to factor 1681.

- We can check if 1681 is a perfect square.
- The square root of 1681 is 41, so:
\[ 1681 = 41^2 \]

Thus, we can rewrite the factorization of 210125 as:
\[ 210125 = 5^3 \times 41^2 \]

### Step 3: Analyze the powers of the prime factors
For a number to be a perfect cube, all the prime factors must have exponents that are multiples of 3.

- In our factorization:
- The power of 5 is 3 (which is already a multiple of 3).
- The power of 41 is 2 (which is not a multiple of 3).

### Step 4: Determine the smallest number to multiply
To make the power of 41 a multiple of 3, we need to increase it from 2 to 3.

- We can do this by multiplying by \( 41^{(3 - 2)} = 41^1 \).

### Conclusion
Thus, the smallest number by which 210125 must be multiplied to make it a perfect cube is:
\[ \text{Required number} = 41 \]

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