Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 305 / 1 / viii / Solution

First, electroweak gauge invariance places left-handed quarks in $SU(2{)}_L$ doublets, pairing one charge-$2/3$ flavor with one charge-$-1/3$ flavor in every generation. Second, cancellation of Standard Model gauge anomalies ties each colored quark doublet and its singlets to a lepton doublet and its singlet; removing either member destroys the hypercharge-anomaly cancellations. Equivalently, the weak charged current and anomaly-free chiral spectrum organize quarks into complete up-down generations, so the number of quark flavors is even.