Can All Linear Diophantine Equations Be Solved?

[Guest2]

Decide which of the following equations are true or false. If false, explain/provide a counterexample.

(a) If $a, b \in \mathbb{Z}$ are relatively prime, then $ax+by = N$ has integer solutions for any integer $N$.
(b) The equation $70x+42y = 1409$ has integer solutions
(c) The equation $70x+42y = 1428$ has integer solutions
(d) The equation $2016x+4031y = 2014201520162017$ has integer solutions.

I think (a) is true since relativity prime means $\gcd(a, b) = 1$ and $1|N$.

 

[Guest2]

(b) $\gcd(70, 42) = 14$ which does not divide the RHS, so it has integer no solutions.
(c) $\gcd(70, 42) = 14$ which divides the RHS, so it has integer solutions.
(d) $\gcd(2016,4031) = 1$ which clearly divides the RHS so it has integer solutions.

Could someone please verify whether this is correct?

 

[kaliprasad]

(b) $\gcd(70, 42) = 14$ which does not divide the RHS, so it has integer no solutions.
(c) $\gcd(70, 42) = 14$ which divides the RHS, so it has integer solutions.
(d) $\gcd(2016,4031) = 1$ which clearly divides the RHS so it has integer solutions.

Could someone please verify whether this is correct?

all including for (a) specified in 1st post correct

 

[Guest2]

all including for (a) specified in 1st post correct

Thanks. Is there a quicker way to do these questions, perhaps using something like modular arithmetic?

 

[I like Serena]

Thanks. Is there a quicker way to do these questions, perhaps using something like modular arithmetic?

(b) can be done quicker, for instance mod 2.
More specifically, the left hand side is even, while the right hand side is odd.