GCD(kj, n) = 1 as kj and n don't share common prime factors.
kj=qn + r for some q
let's construct linear combinations
ks + nt = 1
ju + nv = 1
Multiplying left-hand and right-hand sides
(kj) su + ksnv + ntju + ntnv = 1
(qn+r) su + ksnv + ntju + ntnv = 1
rsu + qnsu + ksnv + ntju + ntnv = 1
r(su) + n(qsu + ksv + tju + tnv) = 1
This shows that there exists such linear combination of r and n that is equal to 1.
Proof completed.
Is it sound and rigorous proof?