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The parallelogram bounded by the lines \(y=ax+c, y=ax+d, y=bx+c,\) and \(y=bx+d\) has area 18. The parallelogram bounded by the lines \(y=ax+c,y=ax-d, y=bx+c, \) and \(y=bx-d\) has area 72. If \(a, b, c,\) and \(d\) are integers, what is the smallest value of \(a \times b \times c+d-2\) ? (i already know the answer is 16, but i don't know how the answer is 16. an explanation would be great.)


Thank you very much!


 Nov 23, 2018

Hi CoolStuffYT     laugh

I did change it a bit, I assumed that a,b,c and d were all POSITIVE integers.                                                                                                                                                                              


I've spent considerable time playing with it and the smallest i have come up with is 28.


That is with  a=1, b=3, c=9  and d=3


It is easy to determine that C=3D

so C must be a multiple of 3


Then I did determine that



the smallest solution for d would be if b-a=2

so b=2+a


So then I said, maybe a=1, b=3, d=3 which makes c=9

\(abc+d-2\\ =1*3*9+3-2\\ =28\)


I am not claiming this is the smallest solution though.

You say the smallest is 16.

Do you have the values of a b c or d to get this solution?  Maybe some of them are negative?

 Dec 1, 2018
edited by Melody  Dec 1, 2018

Ahr, the light dawns,

I think with a and b, one should be positive and the other negative.


a=-1, b=1 d=3, c=9


\(abc+d-2\\ =-1*1*9+9-2\\ =-9+9-2\\ =-2\)


-2 is smaller than your 16.      cheeky


I think all the criteria are met.


There are probably other ones that are smaller still     surprise

 Dec 1, 2018

Here is the diagram for the last answer.


Oh and here is the Desmos graph if you want to play with it:





Melody  Dec 1, 2018


However, the site that I found this question still says 16, I'm not sure why. I think your explanation makes more sense! wink

CoolStuffYT  Dec 2, 2018

hi CoolStuff,

It is not really a matter of making more sense, I am not saying my answer is smallest possible but

I think my answer is correct, in that if all these pronumerals have to be integers, and not necessarily positive integers then 

-2 is a solution to that expession.  Obviously -2 is smaller than 16.


Are you sure you have copied the question exactly?


Have you checked for yourself that my answer works?     

Maybe it doesn't, you should check that it meets all the criterion, I could have made an error.


Did they provide any working for their answer of 16?

Melody  Dec 2, 2018

No, it's a site where only the answer is shown with no explanation.

CoolStuffYT  Dec 3, 2018

It seems to me that the question as stated is incomplete - see my analysis below.

The only way I can see that 16 is the smallest value for abc+d-2 is if a,b,c and d must all be positive integers.


 Dec 2, 2018
edited by Alan  Dec 2, 2018

Well, Alan, thanks anyway. Maybe I missed the positive integers part.

CoolStuffYT  Dec 3, 2018

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