MathDB

au+bv+cw = 0, a = nw- pv, b = pu-mw, c = mv-nu

Source: Romania IMO TST 1989 2.2

February 17, 2020

system of equationsalgebranumber theorycoprime

Problem Statement

Let

a,b,c

be coprime nonzero integers. Prove that for any coprime integers

u,v,w

with

au+bv+cw = 0

there exist integers

m,n, p

such that

\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}