# Why do We Fall? To Rise Again!

$$a$$ and $$b$$ are positive integers such that $$a \ge b$$ and the expression below is an integer: $\frac{a+1}{b}+\frac{b+1}{a}.$ Find the sum of all values of $$a$$ less than $$1000$$ that satisfy the above conditions.

Inspiration: [Problem #1, Spain Mathematical Olympiad, 1996]

$$a$$ and $$b$$ are positive integers such that $\frac{a+1}{b}+\frac{b+1}{a}$ is an integer. Show that the greatest common divisor of $$a$$ and $$b$$ is not greater than $$\sqrt{a+b}$$.

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