If x is a positive integer less than 100, how many values of x make sqrt(1 + 2 + 3 + 4 + x) an integer?
Think of all the perfect squares under 100.
There's 1, 2, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
Now, we know that \(\sqrt{10 + x} = \text{Some integer}\), right?
Because x has to be positive, that rules out 1, 2, 4, and 9 to be the square.
It could be 16, 25, 36, 49, 64, 81, and 100.
That is 7 different numbers, which means 7 different values of x!
