How many pairs of (not necessarily positive) integers satisfy the equation $2xy = 6x + 7y$?

Question

How many pairs of (not necessarily positive) integers satisfy the equation $2xy = 6x + 7y$?

I got this as a practice problem in my book, and I've been pretty stuck on how to find all pairs. I've managed to brute force a few solutions, but I don't really see a pattern. And I'm not sure how to tell once I've found every possible integer solution. Also, I've recently learned that this falls under the Diophantine Equation category, and I know a few identities from that. One identity I know is Simon's Favorite Factoring Trick, but that doesn't seem to work here given the $2$ as the coefficient of the $xy$ term. Any ideas how I can proceed from here, and how I can generally solve this kind of equation?

have you used parity arguments at all ? – Roddy MacPhee Mar 29 '21 at 23:14 — Roddy MacPhee, Mar 29 '21 at 23:14

saulspatz · Accepted Answer · 2021-03-29T23:27:54.067

4

Can't you use SFFT to write $$(2x-7)(y-3)-21=0$$ and then consider the factorizations of $21$?

edited Mar 29 '21 at 23:27

answered Mar 29 '21 at 23:21

saulspatz

53,131

score 0 · Answer 2 · answered Mar 29 '21 at 23:47

0

If you had $2xy = 13x+29y,$ both odd, you could still continue with $$4xy-26x-58y=0, $$ $$ (2x-29)(2y -13) - 377 = 0,$$ $$ (2x-29)(2y -13) = 377 $$

answered Mar 29 '21 at 23:47

Will Jagy

139,541

How many pairs of (not necessarily positive) integers satisfy the equation $2xy = 6x + 7y$?

2 Answers2

Linked