Which smallest number must be added to 710 so that the sum is a perfect cube ?

Options

A. 29
B. 19
C. 11
D. 21

Answer

To solve the problem of finding the smallest number that must be added to 710 so that the sum is a perfect cube, we can follow these steps:

### Step 1: Identify the nearest perfect cube greater than 710.
We need to find the smallest perfect cube that is greater than 710.

### Step 2: Calculate the cube of integers starting from 8.
- \(8^3 = 512\) (not greater than 710)
- \(9^3 = 729\) (this is greater than 710)

### Step 3: Determine the difference between the perfect cube and 710.
Now that we have identified that \(9^3 = 729\) is the nearest perfect cube greater than 710, we can calculate the difference:
\[ 729 - 710 = 19 \]

### Step 4: Conclusion
The smallest number that must be added to 710 to make it a perfect cube is 19.

### Final Answer:
**19**

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