If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 131$, what is the minimal possible value of $|a - b|$?

BRAlNBOLT Jul 11, 2024

#1**+1 **

Let's use a handy trick to solve this problem. We first have

\(ab -3a + 4b = 131 \)

Now, let's take the coefficients of a and b, which are -3 and 4. We multiply them and take the product and add it to both sides. We get

\(ab - 3a + 4b + (4 * - 3) = 131 + (4 * -3) \)

Now, we simplify and factor the left side of the equation. We get that

\(ab - 3a + 4b - 12 = 119 \\ (a + 4) ( b - 3) = 119 \)

Now, let's focus on the factors of 119. We have

\(1 , 7 , 17, 119\)

Clearly, 7 and 17 will get us the minimal value, so plugging that in, we have

\(( 13 + 4) ( 10 - 3) → |a - b | = |13 - 10 | = 3 \)

so 3 is our final answer.

Thanks! :)

NotThatSmart Jul 11, 2024