"Let $n = 24613$, $e = 18041$, $m = 15678$, and $c \equiv m^e \equiv 14304$ (mod $n$).
The element $c$ has order $450$ in the group ($\mathbb{Z}/n\mathbb{Z}$)$^{×}$.
Use this to show that $11$ is the smallest positive integer value of $f$, such that $c^f \equiv m$ (mod $n$)"
Clearly, $14304^{11}\equiv 15678$ (mod $24613$), but I don't understand how to use the order of $c$ to show $11$ is the SMALLEST positive integer value of $f$, without just checking $f = 1,2,3,...,11$
Suppose, an integer $a$ with $0<a<11$ satisfies $$14304^a\equiv 15678\mod 24613$$
Then, we would have $$14304^{11-a}\equiv 1\mod 24613$$ because $15678$ is invertible in $\mathbb Z_{24613}$ , hence the order of $14304$ modulo $24613$ would be smaller than $11$, which contradicts that it is $450$.