diophantine equations theory

Theorem: If a1x1 + a2x2 + .... an.xn = K then this Diophantine equation has a solution if d | K
where d = gcd(a1,a2...an).

We can note the similarities with Bezout's lemma here.

Practice: Which of these Diophantine equations have integer solutions?

1. 21x + 15y = 17
2. 7x + 17y = 27

Theorem: If ax + by = c and d | c, where d = gcd(a,b), then this Diophantine equation has infinitely many solutions, and the solutions are of the form

x = x0 + (b/d).k,
y = y0 - (a/d).k,

where k is integer and (x_0, y_0) are particular solutions of (ax + by = c); and x0, y0 are integers.