Find the general solution in integer

Question

I tried this but my friend has a different answer

Did I make any mistakes?

Answer 1

You want to solve $$41x-17y=8.$$ This is equivalent to solving $$41x \equiv 8 \mod 17.$$ As $41 \equiv 7$ and $7^{-1} \equiv 5 \mod 17$ we get

$$7x \equiv 8 \mod 17$$ $$x \equiv 5\cdot 8 \mod 17$$ $$x \equiv 6 \mod 17.$$

This means $x-6 = 17k$ for integer $t$ or $$x=17t+6.$$ It is clear that $x$ is positive so long as $t \geq 0.$ We can plug to get $$41(17t+6)-17y=8$$ which simplifies to $$41t+14=y.$$ This shows us that for every possible $r\geq 0$ we have a positive $y$.

Notice that there is more than one way to write this answer. I could have instead taken $x=17t+23$ for $t \geq -1$ or $x=17t-11$ for $t \geq 1$.