Existence of integer solution to 63x+70y+15z=2010

Question

I have an equation $63x+70y+15z=2010$. The question asks me to conclude whether it has an integral solution or not? Any help on how to proceed?

Answer 1

Solving the equation certainly gives an affirmative answer.

$$ z=\frac{-21x}{5}+\frac{-14y}{3}+134, $$ hence the solution in integers is $x=5n$, $y=3m$, $z=-21n-14m+134$, where $n$, $m\in\mathbb{Z}$.

Answer 2

The gcd of the coefficients is $1$. In particular, it divides $2010$, so there is an integer solution.

If we want to look more closely, we can see that for example we can take $x=0$. The equation $70y+15z=2010$ has a solution, since the gcd of $70$ and $15$ divides $2010$.