Finding the Smallest Number to Add to 594 for a Perfect Square

The problem asks us to find the minimum positive integer that needs to be added to 594 to make the resulting sum a perfect square. A perfect square is an integer that is the square of another integer (e.g., 9 is a perfect square because $3^2 = 9$).

Step 1: Estimate the Square Root

First, let's find the approximate square root of the given number, 594.

We can estimate or use a calculator:

$ \sqrt{594} \approx 24.37 $

This tells us that 594 lies between the squares of two consecutive integers.

Step 2: Identify Nearby Perfect Squares

Since the square root of 594 is approximately 24.37, the integer part is 24. Let's find the squares of the integers around 24.37:

• The square of 24 is: $ 24^2 = 576 $
• The square of the next integer, 25, is: $ 25^2 = 625 $

We can see that 594 is greater than $24^2$ (576) and less than $25^2$ (625).

Step 3: Determine the Target Perfect Square

We want to add a number to 594 to reach the *next* perfect square. The perfect square immediately following 594 is $25^2$, which is 625.

Step 4: Calculate the Number to Add

To find the smallest number that should be added to 594, we calculate the difference between the target perfect square (625) and the given number (594).

Number to add = Target Perfect Square - Given Number

Number to add = $ 625 - 594 $

Number to add = $ 31 $

Conclusion

Therefore, the smallest number that should be added to 594 so that the sum is a perfect square is 31. Adding 31 to 594 gives $594 + 31 = 625$, which is the perfect square $25^2$.