If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 137$, what is the minimal possible value of $|a - b|$?
I want to find the minimal possible value of $|a - b|$, which is the absolute difference between $a$ and $b$. One way to approach this problem is to try to factor the equation $ab - 3a + 4b = 137$ and see if I can find some restrictions on $a$ and $b$. I notice that $ab - 3a + 4b$ can be written as $a(b - 3) + 4(b - 3) + 12$. This suggests that I can factor out $(b - 3)$ from the equation and get $(a + 4)(b - 3) = 125$. Now I have a product of two factors that equals 125. Since $a$ and $b$ are positive integers, I can look for pairs of factors of 125 that have the form $a + 4$ and $b - 3$. The factors of 125 are 1, 5, 25, and 125. If $a + 4 = 1$, then $a = -3$, which is not positive, so I can eliminate this possibility. If $a + 4 = 5$, then $a = 1$ and $b - 3 = 25$, so $b = 28$. If $a + 4 = 25$, then $a = 21$ and $b - 3 = 5$, so $b = 8$. If $a + 4 = 125$, then $a = 121$ and $b - 3 = 1$, so $b = 4$. These are the only possible pairs of values for $a$ and $b$ that satisfy the equation. Now I need to compare the absolute differences $|a - b|$ for each pair and find the minimal value. For $a = 1$ and $b = 28$, $|a - b| = 27$. For $a = 21$ and $b = 8$, $|a - b| = 13$. For $a = 121$ and $b = 4$, $|a - b| = 117$. The smallest of these values is 13, so the minimal possible value of $|a - b|$ is 13. 