Let a,b,c be positive integers such that ab+bc-ca=0 and a-c=523. Find b.

Question

ganeshie8 · Answer

what have you tried ?

anonymous · Answer

so this is a diff eq problem?

anonymous · Answer

set c= 523 plug them all in and try to find the answer

mathmate · Answer

@hzer I read $a-c=523$

anonymous · Answer

ok then what

anonymous · Answer

well ok that is ok but also if you plug all of them in it will not give you the right one don't you think so mathmate

hartnn · Answer

this could be a start :
$\Large \dfrac{1}{b}=\dfrac{1}{a}+\dfrac{1}{c}$

hartnn · Answer

b= c(c+523)/(2c+523)

hartnn · Answer

for b to be a positive integer 
2c+523 should divide  c (c+523)

mathmate · Answer

@hartnn
I solved  for c in c(c+523)/(2c+523)$\in$ N  [but by brute force]:
a=137026=2*131*523
b=68382=2*3^2*29*131
c=136503=3^2*29*523
and 
1/b=1/a+1/c
ab+bc-ca=0
...and there could be other solutions.

I wish I knew how to solve for c "mathematically".