Leading-residue obstruction to partition regularity ($c(x)=p^{-v_p(x)}x\mathrm{mod}p$)

For nonzero integer coefficients $a_i$, choose a prime number $p>{\sum }_i|a_i|$ and colour each positive integer by its nonzero leading residue after removing its power of $p$. In a monochromatic zero-sum solution, divide by the least occurring power of $p$. Reduction modulo $p$ then makes the sum of the coefficients attached to the least-valuation variables vanish modulo $p$. Its absolute value is less than $p$, so that sum vanishes as an integer. This gives the necessity direction of the one-equation partition regularity criterion without using the general Rado theorem.