What is the least number which is added to 680621 to make the sum a perfect square ?

Question

Accepted Answer

To determine the least number that needs to be added to 680621 to make it a perfect square, we need to find the smallest integer $ n $ such that $ 680621 + n $ is a perfect square.

1. **Calculate the square root of 680621:**
   - The approximate square root of 680621 is 824.87. This means that 680621 is slightly less than the square of 825.

2. **Calculate the square of 825:**
   - $ 825^2 = 680625 $.

3. **Determine the difference:**
   - The difference between 680625 and 680621 is $ 680625 - 680621 = 4 $.

Therefore, adding 4 to 680621 results in 680625, which is a perfect square ($ 825^2 $). This confirms that the least number to be added is 4, making Option 3 the correct answer.

**Explanation for why other options are incorrect:**

- **Option 1 (5):** Adding 5 to 680621 gives 680626, which is not a perfect square. The square root of 680626 is approximately 825.0006, which is not an integer.

- **Option 2 (6):** Adding 6 to 680621 results in 680627, which is not a perfect square. The square root of 680627 is approximately 825.0012, which is not an integer.

- **Option 4 (8):** Adding 8 to 680621 gives 680629, which is not a perfect square. The square root of 680629 is approximately 825.0024, which is not an integer.

Thus, the only option that results in a perfect square is Option 3, where adding 4 to 680621 results in 680625, a perfect square.