If I know that a and b are integers, how would I prove that a=b in the equation ab=a+b?

Question

cwrw238 · Answer

i dont understand that question - the only values i can think of to fot satisfy it is if a and b wre both = 2

kainui · Answer

Or 0, I suppose I'm just curious how to come to this conclusion logically rather than just guessing.

cwrw238 · Answer

oh ok

anonymous · Answer

see if this helps or not$$b=\frac{a}{a-1}=1+\frac{1}{a-1}$$from here show that$$a=b=2$$

anonymous · Answer

and or$$a=b=0$$

raden · Answer

ab-a=b
a(b-1)=b
a=b/(b-1)
a would be integer, satisfied if numerator (b) is 0 or denominator (b-1) is 1

kainui · Answer

Ah thank you very much I see. By making it in terms of fractions we eliminate the impossible solutions by seeing fairly obviously which values will give non-integer answers. Cool.

raden · Answer

thanks for medal, my teacher @mukushla :)

anonymous · Answer

Oh...man :) ur very welcome my friend :)

raden · Answer

:)

kainui · Answer

If you're having fun, can we take this a couple steps further and look at how to show abc=a+b+c as all integers can only allow 0 or 1,2,3 as answers (in 3! ways). Since the extra variables are involved it gets a little trickier.

Then I noticed I could extend these rules to n number of variables so that abcde=a+b+c+d+e would have answers of 0 or 1,1,1,2,5. But it might possibly have more. Anyone interested in playing around with this with me for fun?

anonymous · Answer

sure.. i'll look at this later...

kainui · Answer

Haha alright. If you know any websites that talk about this or what this is called if it has a name, that would be extrodinarily helpful.