= Solution
The <row orthogonality relations for a character table> state that for irreducible characters $\chi,\psi$,
$$
\boxed{\sum_{g\in G}\chi(g)\overline{\psi(g)}
=|G|\delta_{\chi\psi}.}
$$
Equivalently, if $c$ runs through representatives of the conjugacy classes $C$,
$$
\sum_C|C|\chi(c)\overline{\psi(c)}
=|G|\delta_{\chi\psi}.
$$
Fix an irreducible $\chi$. The sum of the elements in a conjugacy class $C$ is central in $\mathbb CG$, so <Schur lemma> says that it acts in the representation affording $\chi$ by the scalar
$$
\omega_\chi(C)=\frac{|C|\chi(c)}{\chi(1)}.
$$
This <central character value of a conjugacy-class sum> is an <algebraic integer>: the class sum acts by a matrix with integer entries on the <regular representation>, and $\omega_\chi(C)$ is one of its eigenvalues. Also $\overline{\chi(c)}$ is an algebraic integer because character values are sums of roots of unity.
Row orthogonality with $\psi=\chi$ now gives
$$
\frac{|G|}{\chi(1)}
=\sum_C
\frac{|C|\chi(c)}{\chi(1)}\overline{\chi(c)}
=\sum_C\omega_\chi(C)\overline{\chi(c)}.
$$
The right-hand side is an algebraic integer. The left-hand side is rational, and every rational algebraic integer is an integer. Therefore
$$
\boxed{\chi(1)\mid|G|.}
$$
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