 Solution (source code)

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= Solution

In the integer <basis> $u,v,e_3$, direct substitution gives $w=-7u+5v+18e_3$. Therefore
$$
M=\mathbb Zu\oplus\mathbb Zv\oplus18\mathbb Ze_3,
\qquad
\boxed{L/M\cong\mathbb Z/18\mathbb Z.}
$$
The first two <basis> coordinates disappear in the quotient and the third is taken modulo eighteen. Equivalently the <Smith normal form> has diagonal entries $1,1,18$.

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