= Solution
The cusps are the $\Gamma_1(p)$-orbits of primitive columns $(a,c)^T$, with simultaneous negation representing the same point of $\mathbb P^1(\mathbb Q)$. For the <Gamma 1 congruence subgroup>, the standard primitive-vector classification separates the orbits according to $d=\gcd(c,p)$. For each divisor $d\mid p$, the two surviving unit coordinates give
$$
\frac12\varphi(d)\varphi(p/d)
$$
orbits when $p>2$. Hence an odd prime gives
$$
\#\bigl(\Gamma_1(p)\backslash\mathbb P^1(\mathbb Q)\bigr)
=\frac12\sum_{d\mid p}\varphi(d)\varphi(p/d)
=p-1.
$$
For $p=2$, simultaneous negation is already trivial modulo $2$, so the division by two does not apply. In that case $\Gamma_1(2)=\Gamma_0(2)$ has two cusps, represented by infinity and zero. Thus the <Number of cusps of Gamma 1 of prime level> is
$$
\begin{cases}
2,&p=2,\\
p-1,&p\text{ odd}.
\end{cases}
$$
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