Which are the integer solutions of this function?

Question

Consider

$215\cdot x - 1763\cdot y=z$ with x, y, z integer.

Which form must have x and y?

I tried something like $x=41\cdot n$ and $y=5\cdot n$ or $x=9+41\cdot n$ and $y=1+5\cdot n$, are there other forms?

Answer 1

$z=43(5x-41y)$

So, $z$ must be multiple of $43, z=43w$(say)

Like Solving a Linear Congruence

$$41-8\cdot5=1$$

$$5x-41y=w(41-8\cdot5)\iff5(x+8w)=41(w+y)$$

$\dfrac{41(y+w)}5=x+8w$ which is an integer

$\implies5$ must divide $41(y+w)$

As $(5,41)=1,5$ must divide $y+w,$ so $y+w$ can be set as $5t\iff y=5t-w$

$x+8w=\dfrac{41(y+w)}5=41t$

Answer 2

This is a linear diophantine equation of the form $ax +by = cz$. There is a solution only when $gcd(a, b)|c$ (the greater common divisor between $a$ and $b$ must divides $c$), and when this condition is matched there is an infinite number of solutions of the form: $(x_0 + k\frac{b}{gcd(a, b)}, y_0 - k\frac{a}{gcd(a, b)})$ where $(x_0, y_0)$ is a particular solution for $(x, y)$ and $k$ ranges over $\mathbb{Z}$.

In this case $c=1$ hence $a=215$ and $b=−1763$ must be coprimes in order to have e solution. We can calculate $gcd(215, −1763)$ with the eculidean algorithm.

$1763=8 \times 215 + 43 \\ 215=5\times 43$

Hence $gcd(215, −1763)=gcd(1763, 215)=43$ and your equation has not integer solutions.