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avatar+183 

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
avatar
+1

I get a=44  and  b =13

So that: abs[44 - 13] =31

 Oct 19, 2018
 #2
avatar+316 
+2

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
avatar+183 
+1

Thank y'all both! :)

 Oct 19, 2018
 #4
avatar+98044 
+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

 

 

cool cool cool

 Oct 19, 2018

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