If $a > 0$,$b>0$, and $\frac{1}{a} + \frac{1}{b}$ is an integer, prove that $a=b$. And show that $a = 1$ or $2$ If $a$ and $b$ are positive integers, and $\frac{1}{a} + \frac{1}{b}$ is an integer, prove that $a=b$. And show that $a = 1$ or $2$
-I played around with numbers and the conditions and it seems that it's common sense that if $a = b$ then $a = b = 1$ or $2$, For example:
If $a = b = 4$, then $\frac{1}{a} + \frac{1}{b}$ would not be an integer.
-Any help is appreciated

If $\frac1a + \frac1b = \frac{a+b}{ab}$ is an integer, then $a$ and $b$ both divide $a+b$. This means that $a$ divides $(a+b)-a=b$ and that $b$ divides $(a+b)-b=a$. So $a$ and $b$ divide each other, whence they must be equal.
Now if $\frac1a + \frac1b = \frac2a$ is an integer, then $a$ must divide $2$. So $a=1$ or $a=2$.
