What is the Minimum Number to Add to Make a Number a Perfect Square?

One common question in mathematics is finding the minimum number that should be added to a given number to make it a perfect square. In this article, we will explore how to determine the answer using the example of 280. We will go through the step-by-step process of finding the nearest perfect squares and the smallest number to add to make 280 a perfect square. Additionally, we will extend our discussion to consider negative numbers, as well as the smallest positive integer that can achieve the same result.

Example: Making 280 a Perfect Square

To find the minimum number that should be added to 280 to make it a perfect square, we start by calculating the square root of 280.

sqrt{280} approx 16.733

The next whole number after 16.733 is 17. We then calculate the square of 17:

17^2 289

Next, we find the difference between 289 and 280 to determine the minimum number to add:

289 - 280 9

Therefore, the minimum number that should be added to 280 to make it a perfect square is 9.

Finding the Least Positive Integer for Higher Numbers

Let's now consider a higher number, such as 280567. We want to find the smallest positive integer that can be added to 280567 to make it a perfect square.

The square root of 280567 is approximately:

sqrt{280567} approx 529.686

Rounding up to the nearest whole number gives us 530. Squaring 530:

530^2 280900

We can now calculate the difference between 280900 and 280567:

280900 - 280567 333

Thus, the smallest positive integer that should be added to 280567 to make it a perfect square is 333.

Considering Negative Numbers

It is also important to note that we can use negative numbers to achieve a perfect square. For instance, the square of 1 is:

1^2 1

This means that:

280900 - 280899 1^2

Therefore, -280899 is the smallest negative integer that can be added to 280900 to make it a perfect square.

Conclusion

By following these steps, we have determined that the minimum number that should be added to 280 to make it a perfect square is 9, the smallest positive integer that can be added to 280567 to achieve the same result is 333, and the smallest negative integer that can achieve this is -280899. This exploration demonstrates the importance of considering both positive and negative numbers in mathematical calculations involving perfect squares.