I have to prove that any number that divided by 5 gives a remainder of 1 and divided by 7 a remainder of 2, also gives a remainder of 16 when divided by 35, using a diophantine equation.
So first I made the following proposition: $(n=5q_1+1 \land n=7q_2+2) \implies 16=n-35q_3$ and started by working with the hipothesis hoping to reach the conclusion.
Since $(n=5q_1+1 \land n=7q_2+2)$, then $5q_1+1=7q_2+2$, thus I can also say that $5q_1-7q_2=1$.
Here I have my diophantine equation. From now on, $x=q_1$ and $y=q_2$ so I can have $5x-7y=1$
$GCD(5, -7)=1$, and a linear combination for $5p-7q=GCD(5, -7)$ is $p=3$ and $q=2$. Since $GCD(5, -7) = c = 1$, $x_0=(c*p)/d=3$ and $y_0=(c*q)/d=2$
But now that I have this, I don't have any clue how to continue. How should I limit the value of $k$?