My opinion: powerful and effective problems will increase interest and motivation.
For example, We can prove Wilson's Theorem with using Fermat's little theorem. Therefore (If preferred) we can take Wilson's theorem an appication of Fermat's. More specifically,
When $p=2$ easily $(2-1)!\equiv -1\pmod{2}$. Let $p>2$ be a odd prime. If we solve
$$x^{p-1} - 1 \equiv 0 \pmod{p} \tag{1}$$
by Fermat's theorem $x=1,2,\dots , p-1$ satisfy $(1)$. Each root is a factor of $x^{p-1} - 1$ polynomial and $$ x^{p-1} - 1 \equiv (x-1)(x-2)\cdots (x-(p-1))\pmod{p} \tag{2}$$
If we put $x=0$ in $(2)$ then $1\cdot 2 \cdots (p-1) \equiv -1 \pmod{p}$
Moreover, if we find direct applications/examples of Wilson's Theorem, (I hope that) they may increase the motivating. I can give two Wilson's Theorem applications:
1) $\dfrac{10!}{13}+x$ is a positive integer then, what is mimimum positive real value of $x$?
2) If $16!$ divided by $323$, what is the remainder? (You may give a hint: $323= 17\cdot 19$.)