Plot the graph $24x + 120y= gcd(24, 120)$. By using this graph or otherwise, find two integer solutions to this equation:
$$24x + 120y = gcd (24, 120)$$
The $gcd(24, 120)=24$, so we plot (desmos graph).
$$24x + 120y = 24$$
We can see immediately that $x=1, y=0$ looks like a solution, and checking in the equation confirms this.
The equation can be divided through by the gcd to give
$$x + 5y = 1$$
Re-arranging gives
$$y = \frac{1 - x}{5}$$
In order for $y$ to be an integer, $1-x$ must be a multiple of 5. The choice $x=6$ works, giving $y=-1$.
So the two integer solutions are:
- $x = 1, y=0$
- $x = 6, y=-1$