TITLE: Proving there are no integers $a, b, c$ satisfying $12a + 18b + 27c = 227$
QUESTION [6 upvotes]: Given $12a + 18b + 27c = 227$, how can we prove that $a, b, c$ can never be integers? I don't have many ideas. Can someone give me some ideas?

REPLY [4 votes]: Generally, if $\rm\:i,j,k,n\in \Bbb Z\:$ then $\rm\: i\,a + j\,b + k\,c = n\:$ has a solution $\rm\:a,b,c\in \Bbb Z\iff gcd(a,b,c)\mid n.\:$ The necessity $\rm\:(\Rightarrow)\:$ is clear, and the sufficiency $(\Leftarrow)$ follows from Bezout, i.e. integers of form $\rm\:i\,a + j\,b + k\,c\:$ are closed under subtraction, so closed under remainder, so closed under gcd.
Your example fails the criterion: $\rm\: (12,18,27) = 3(4,6,9)=3,\:$ but $\rm\:mod\ 3\!:\ 227\equiv 2\!+\!2\!+\!7\equiv 2\not\equiv 0.$