Solution

Suppose $\text{ord}_n b = 3$. How many positive integer solutions does $b^x \equiv 1 \pmod{n}$ have for $x$ that are less than $26$?



Recall that $b^x \equiv 1 \pmod{n}$ if and only if $\textrm{ord}_n b \mid x$.

As there are only $8$ positive integers between $1$ and $26$ that $3$ divides evenly (i.e., $3$, $6$, $9$, $12$, $15$, $18$, $21$, and $24$), there must be exactly $8$ solutions for $x$.