$7x+5y=12$, so why does its solution will be of the format $(1+5n,1-7n)$

Question

Question

Consider all ordered pairs of integers $(x,y)$ such that $$\frac{5}{x}+\frac{7}{y} = \frac{12}{xy}.$$ The smallest positive integer value of $x$ in these ordered pairs is 1, since $x=y=1$ satisfies the equation. What is the second smallest positive integer value of $x$ in these ordered pairs?

If I rearrange the equation I get $7x+5y=12$. I have been told that $5$ and $7$ are relatively coprime therefore solution will be of the format $(1+5n,1-7n)$

The bold Part in above argument is what I don't understand.

Only integer values of $x$ make both $5x$ and $7x$ be integers — WW1, Mar 01 '23 at 02:52
Dupe of Find all integer solutions to $7595x + 1023y=124$. In particular see the general solution method in this answer — Bill Dubuque, Mar 02 '23 at 07:03
Here's another dupe, and see here for analogs for differential and difefrence equations (recurrences). — Bill Dubuque, Mar 02 '23 at 07:07
@BillDubuque thanks for your reply I found it helpful, can you guide me how do you remember such results. — Ayush Upadhyay, Mar 02 '23 at 10:12
The idea I describe in elementary langauge about the form of the general solution is a basic result of linear algebra / modules (which will be clarified when rephrased more structurally in terms of images and kernels and the associated isomorphism theorem). Thanks for the typo correction. — Bill Dubuque, Mar 02 '23 at 10:17

score 0 · Accepted Answer · edited Mar 01 '23 at 03:12

0

Hint: If a prime $p$ divides $ab$ then either $p$ divides $a$ or $p$ divides $b$.

Now use it in the relation $7(x-1)=5(1-y)$.

edited Mar 01 '23 at 03:12

Clyde Kertzer

2,904

answered Mar 01 '23 at 02:37

Soumik Mukherjee

525

can you guide me how develop thinking like that, any book to read or refer me to any resource. – Ayush Upadhyay Mar 01 '23 at 02:44
I think it will be n only as if fix one variable other will also be fixed as answer is integer. – Ayush Upadhyay Mar 01 '23 at 02:48
Yes n will be fixed – Soumik Mukherjee Mar 01 '23 at 03:05
Solving Mathematical Problems by Terence Tao is a great book to develop mathematical thinking – Soumik Mukherjee Mar 01 '23 at 03:06

$7x+5y=12$, so why does its solution will be of the format $(1+5n,1-7n)$

1 Answers1