Determine whether the following equations have integer solutions. If they do have solutions, find the general solution:
(a) $2x + 4y= 1$
(b) $48x + 56y= 32$
(c) $54x + 180y= −72$
(a) Here $\gcd(2,4) = 4$, which does not divide 1. Therefore the equation does not have integer solutions.
(b) Here $\gcd(48,56)=8$, which divides 32. Therefore the equation does have integer solutions.
By trial and error we find one solution $x_0=10, y_0=-8$. The general solution is
$$ x = 10 + 7t \quad y =-8 - 6t$$
for any integer $t$.
(c) Here $\gcd(54, 180)=18$, which does divide -72. Therefore the equation does have integer solutions.
By trial and error we find one solution $x_0=2, y_0=-1$. The general solution is
$$ x = 2 + 10t \quad y =-1 - 3t$$
for any integer $t$.