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If a and b are positive integers for which ab-6a+5b=15, what is the minimal possible value of |a-b|?

 Jun 15, 2021
 #1
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The minimal possible value of |a - b| is 3.

 Jun 15, 2021
 #2
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Substituting $b=6$ into $ab-6a+5b=15$ we get the absurdity $30=15$.   Thus $b$ can never assume the value 6.   By algebraic manipulation we can express $a$ in terms of $b$ as $a = (15-5b)/(b-6)$.    Now, $15-5b > 0$ iff $b < 3$ and $b-6 > 0$ iff $b > 6$.   Therefore, $a$ is positive only when $3 < b < 6$.   Among these two plausible values of $b$ (4 and 5), only $b=5$ turns $a$ into a positive integer $a=10$.  Thus, $|a-b|$ has only one value 5 and it must be the minimum.

 Jun 15, 2021

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