= Solution
One generation of left-handed <Standard Model fermions> fits into $\overline{\mathbf5}\oplus\mathbf{10}$ of an <SU(5) grand unified theory>. With <hypercharge> defined by $Q=T_3+Y$, their <restriction of a representation> to $SU(3)_c\times SU(2)_L\times U(1)_Y$ is
$$
\begin{aligned}
\overline{\mathbf5}&\longrightarrow(\overline{\mathbf3},\mathbf1)_{1/3}\oplus(\mathbf1,\mathbf2)_{-1/2}
&&\text{carrying }d^c,L,\\
\mathbf{10}&\longrightarrow(\mathbf3,\mathbf2)_{1/6}\oplus(\overline{\mathbf3},\mathbf1)_{-2/3}\oplus(\mathbf1,\mathbf1)_1
&&\text{carrying }Q,u^c,e^c.
\end{aligned}
$$
Here $Q$ in the multiplet list denotes the left-handed <quark> doublet, and the superscript $c$ labels left-handed <charge-conjugate fields> of the corresponding right-handed <fermions>.
The results of <4/c> therefore give
$$
\boxed{\overline{\mathbf6}\oplus\mathbf{15}\longrightarrow
\underbrace{\overline{\mathbf5}\oplus\mathbf{10}}_{\text{one ordinary generation}}
\oplus\underbrace{\mathbf1\oplus\mathbf5}_{\text{additional multiplets}}.}
$$
Thus the required matter <representations> are present. The extra singlet has <Standard Model representation> $(\mathbf1,\mathbf1)_0$ and can describe the left-handed <charge-conjugate field> of a <right-handed neutrino>. Its <gauge singlet> character permits a <Majorana mass term>; a <Yukawa interaction> with a lepton doublet and a suitable <Higgs field> can then supply neutrino masses. The <branching> alone does not fix those interactions or their scales.
The extra $\mathbf5$ has
$$
\mathbf5\longrightarrow(\mathbf3,\mathbf1)_{-1/3}\oplus(\mathbf1,\mathbf2)_{1/2}.
$$
If it describes <Weyl fermions>, these are additional coloured and electroweak <fermions> in the conjugate <Standard Model representations> to the ordinary $\overline{\mathbf5}$. With an additional $\overline{\mathbf5}$, they can form a <vectorlike gauge spectrum> and acquire invariant masses. Pairing them with the only ordinary $\overline{\mathbf5}$ would also remove that generation's $d^c,L$ fields, so a realistic heavy pairing needs extra matter. If the $\mathbf5$ instead describes a <scalar field>, it has the quantum numbers of the usual $SU(5)$ <Higgs field>: a weak doublet and a coloured triplet. A single fermionic $\mathbf{15}$ does not turn its extra fermions into a scalar <Higgs field> merely by <branching>.
There is a further consistency condition if these are the entire chiral matter content of an $SU(6)$ <gauge theory>. Normalize the <cubic anomaly coefficient> by $A(\mathbf6)=1$. The <dual representation> has $A(\overline{\mathbf6})=-1$, and the <two-index antisymmetric representation> has $A(\mathbf{15})=6-4=2$. Thus one copy of each has nonzero <gauge anomaly>, $A_{\mathrm{total}}=1$. Their inclusion of the desired $SU(5)$ multiplets is therefore not, by itself, a complete consistent chiral unified theory.
For example, <anomaly cancellation> holds for $\mathbf{15}\oplus2\overline{\mathbf6}$, whose <restriction of a representation> is
$$
\boxed{\mathbf{15}\oplus2\overline{\mathbf6}\longrightarrow
(\mathbf{10}\oplus\overline{\mathbf5})\oplus(\mathbf5\oplus\overline{\mathbf5})\oplus2\mathbf1.}
$$
This leaves one ordinary generation, a potentially massive <vectorlike gauge spectrum>, and two possible singlet neutrinos. This anomaly-free matter choice is also used in explicit $SU(6)$ model constructions.<Number of fermion generations from a novel grand unified model by Pritibhajan Byakti and others>[Section 3, equations (14)–(16) and following discussion: the fifteen-dimensional antisymmetric representation has anomaly coefficient 2, and the conjugate fundamental has coefficient −1]{ref} These interpretations require additional choices of matter statistics, <spontaneous symmetry breaking>, and interactions; the <weight systems> determine the multiplet content.
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