Question

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

Answer 1

what have you tried ?

Answer 2

so this is a diff eq problem?

Answer 3

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

Answer 4

@hzer I read \(a-c=523\)

Answer 5

ok then what

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

@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".