Fermat's little theorem and compositeness

Fermat test. If $15$ were prime, then $a^{14}\equiv1\pmod{15}$ for all $a$ coprime to $15$.

Try $a=2$. $2^4=16\equiv1\pmod{15}$, so $2^{14}=2^{12}\cdot2^2=(2^4)^3\cdot4\equiv4\pmod{15}\neq1$.

Conclusion. $2^{14}\not\equiv1\pmod{15}$, so $15$ is composite.