Final answer:
The smallest positive integer that must be added to 2010 to obtain a perfect square is 45.
Step-by-step explanation:
To find the smallest positive integer that must be added to 2010 to obtain a perfect square, we need to consider the properties of perfect squares. A perfect square is a number that can be expressed as the square of an integer. We can use trial and error to find the smallest positive integer that, when added to 2010, results in a perfect square.
Starting with 2010, we can add 1, 2, 3, and so on until we find a perfect square. By doing this, we can determine that adding 45 to 2010 yields a perfect square: 2010 + 45 = 2055, and the square root of 2055 is approximately 45.36.
Therefore, the smallest positive integer that must be added to 2010 to obtain a perfect square is 45.