Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 3 / 14F / iv / Solution

In the integer basis $u,v,e_3$, direct substitution gives $w=-7u+5v+18e_3$. ThereforeThe first two basis coordinates disappear in the quotient and the third is taken modulo eighteen. Equivalently the Smith normal form has diagonal entries $1,1,18$.