Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 305 / 3 / ii

The $[SU(3){]}^{2}U(1)$ condition gives $u+d=2q$, which is even because all hypercharges are integers. The sum $u+d$ and difference $u-d$ have the same parity, so $u-d$ is even. ThereforeSolving the sum and difference equations gives