+283 If a and b are positive integers for which ab-6a+5b=373, what is the minimal possible value of |a-b|?
+343 Perfect will be a=b, because \(\left |a-b \right |=> \left |a-a \right |=0\) so for \(a=b\) we have:
\(a^2 -6a + 5a - 373 =0\) <=> \(a^2 -a - 373 =0\)
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)=> \(x = {1 \pm \sqrt{1-4(1)(-373)} \over 2}\)=>\(x = {1 \pm \sqrt{1493} \over 2}\)=> \(x = {1 \pm ~38.63 \over 2}\)
\(x1=\frac{39.63}{2} = 19.81\)
\(x2=\frac{-37.63}{2}=-18.81\) BUT because is negative rejected
so \(a=20 \) so \(20b-120+5b=373\) => b=19.72 its NOT integers so you try values until a,b integers
Finaly the answer is a=44 and b =13 like Guest says and can see it in this graph if you try values!
https://www.desmos.com/calculator/4fhytbda6x
Hope it helps!
+128475 ab - 6a + 5b = 373
We can write this as
ab - 6a + 5b - 30 = 373 - 30
(a + 5 ) ( b - 6 ) = 373 - 30
(a + 5) (b - 6) = 343
343 as a product of two factors =
1 * 343 or 343 * 1
7 * 49 or 49 * 7
So....we have the following possibilities for a, b
a b
338 7
2 55
44 13
So.... l a - b l minimized = l 44 - 13 l = 31
