+216 If $a$ and $b$ are positive integers for which $ab - 3a + 4b = 131$, what is the minimal possible value of $|a - b|$?
+1950 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! :)
