Introduction:

To find the least number that must be added to 4931 to make it a perfect square, we need to determine the nearest perfect square that is greater than 4931. By finding the difference between this perfect square and 4931, we can determine the least number that needs to be added.

Steps to find the least number:

1. Identify the nearest perfect square:
- The square root of 4931 is approximately 70.27.
- The nearest perfect square to 4931 is 4900, which is equal to (70 * 70).

2. Determine the difference:
- The difference between 4931 and 4900 is 31.

3. Verify if the difference is the least number:
- Now, we need to check if 31 is the least number that needs to be added to make 4931 a perfect square.
- We can do this by calculating the square root of (4931 + 31) and checking if it is an integer.
- The square root of (4931 + 31) is approximately 71.07, which is not an integer.

4. Increment the difference:
- Since 31 is not the least number, we increment the difference by 1 and repeat step 3.
- The new difference is 32.

5. Continue incrementing the difference:
- We repeat step 3 by calculating the square root of (4931 + 32) and checking if it is an integer.
- The square root of (4931 + 32) is approximately 71.09, which is not an integer.

6. Repeat until an integer is obtained:
- We continue incrementing the difference by 1 and repeating step 3 until we find a number where the square root is an integer.
- Eventually, we find that the difference of 49 makes (4931 + 49) a perfect square.

Conclusion:

The least number that must be added to 4931 to make it a perfect square is 49. Adding 49 to 4931 results in 4980, which is the nearest perfect square.