+0

0
540
4

If a and b are positive integers for which ab-6a+5b=373, what is the minimal possible value of |a-b|?

Oct 19, 2018

#1
+1

I get a=44  and  b =13

So that: abs[44 - 13] =31

Oct 19, 2018
#2
+1

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!

Oct 19, 2018
edited by Dimitristhym  Oct 19, 2018
#3
+1

Thank y'all both! :)

Oct 19, 2018
#4
+3

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   Oct 19, 2018