One example are the repunits where $a=1$, $b=1$, and $c=9$. Another example are the numbers of the form 133...331 which are generated by the values $a=4$, $b=7$, and $c=3$. There are lots of other examples. Is there a way to determine for a given value of $a$ which values are always guaranteed to return an integer?
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Did you mean $a=1$ for repunits? – J. W. Tanner Dec 05 '19 at 16:49
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3It works when always $a\cdot10^n\equiv b\pmod c$, which holds for the examples you gave because $a\equiv b$ and $10\equiv 1\pmod c$ – J. W. Tanner Dec 05 '19 at 16:50
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@J.W.Tanner I guess what we are really after is when the values of a,b,c return integer terms which do not always share a set of factors? So in some sense they are in 'lowest terms' and it makes it possible to ask when they are prime... I edited the question to be more restrictive. – Goldbug Dec 05 '19 at 17:04