What is the least positive integar that should be added to 720 so that the sum is a perfect cube ?

Answer

To find the least positive integer that should be added to 720 so that the sum is a perfect cube, we can follow these steps:

### Step 1: Understand Perfect Cubes
Perfect cubes are numbers that can be expressed as \( n^3 \), where \( n \) is a positive integer. The first few perfect cubes are:
- \( 1^3 = 1 \)
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)
- \( 4^3 = 64 \)
- \( 5^3 = 125 \)
- \( 6^3 = 216 \)
- \( 7^3 = 343 \)
- \( 8^3 = 512 \)
- \( 9^3 = 729 \)
- \( 10^3 = 1000 \)

### Step 2: Identify the Perfect Cube Greater than 720
We need to find the smallest perfect cube that is greater than 720. From our list, we see:
- \( 9^3 = 729 \) is the first perfect cube greater than 720.
- \( 10^3 = 1000 \) is too large for our needs.

### Step 3: Calculate the Required Integer
Now, we need to find the least positive integer \( k \) such that:
\[ 720 + k = 729 \]
To find \( k \), we rearrange the equation:
\[ k = 729 - 720 \]
Calculating this gives:
\[ k = 9 \]

### Conclusion
The least positive integer that should be added to 720 to make it a perfect cube is \( \boxed{9} \).

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