<s>[INST] 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
 [/INST] A variant (supposing $a,b$ are natural numbers):
Say $a\le b$. The hypothesis means $ab=a+b\le 2b$, whence $a\le 2$. Furthermore $b\mid a+b$, hence $a+b-b=a$, so that $a=b$.
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