I just saw the next two theorems and I asked myself some questions about it.
Linear Diophantine Equation:
Let a and b be integers with $d=(a,b)$. The equation $ax+by=c$ has:
1) No integral solutions if $c$ is not divisible by $d$
2) Infinity many integral solutions if $c$ is divisible by $d$
The solutions are given by: $x=x_0 + \frac{b}{d}n, y=y_0+\frac{a}{d}n$
Linear congruences:
Let a, b and m be integers with $d=(a,m)$. The congruence $ax\equiv b(\mod m)$ has:
1) No solutions if $c$ is not divisible by $d$
2) d incongruent solutions if $c$ is divisible by $d$
The solutions are given by: $x=x_0 + \frac{m}{d}t, y=y_0+\frac{a}{d}t$
The linear congruence $ax\equiv b(modm)$ is equivalent to $ax-my=b$.
Question 1:
1) Diophantine Equation: $ax+by=c$ and solution $x=x_0 + \frac{b}{d}n, y=y_0+\frac{a}{d}n$
2) Linear congruence: $ax-my=b$ and solution $x=x_0 + \frac{m}{d}t, y=y_0+\frac{a}{d}t$
The only thing that has changed in the solution (2) is that $b$ is replaced by an $m$, but why is that the only change? The $+b$ in $ax+by=c$ has also become an $-m$ in $ax-my=b$?
Question 2:
1) Diophantine Equation: Infinity many integral solutions if $c$ is divisible by $d$
2) Linear congruence: d incongruent solutions if $c$ is divisible by $d$
Why are there first infinity many solutions and in the second case only d?
