We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
540
4
avatar+269 

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+342 
+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
avatar+269 
+1

Thank y'all both! :)

 Oct 19, 2018
 #4
avatar+104686 
+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

33 Online Users

avatar