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Given that $a$ and $b$ are positive integers. and $$f(x) = \frac{ax + b}{a -x + 1}$$ is an integer, what integer values can x have?

If I could only somehow move $x$ from numerator to the denominator I would be able to solve this by factoring the numerator.

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  • $\begingroup$ Top part is called "numerator" and bottom part "denominator". $\endgroup$
    – Gerhard S.
    Commented Nov 11, 2017 at 20:42

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Since $$a-x+1\mid ax+b \;\;\;{\rm and} \;\;\;a-x+1\mid a(a-x+1)$$ we have $$a-x+1\mid (ax+b)+(a^2-ax+a)= a^2+a+b$$ So $x=a-d+1$ where $d$ divides $a^2+a+b$.

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  • $\begingroup$ Can you please link me to some wiki page about the math you used here, I am not familiar with that stick sign $\endgroup$
    – Ilya Gazman
    Commented Nov 11, 2017 at 22:39
  • $\begingroup$ en.wikipedia.org/wiki/Divisor $\endgroup$
    – nonuser
    Commented Nov 14, 2017 at 12:51

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