Proving equivalent the below characterizations of squarefree naturals is quite elementary employing cyclic inferences (assuming one knows the existence and uniqueness of prime factorizations).
Theorem $\ $ The following are equivalent for a natural number $\,q > 1$
$(1)\qquad\quad n^2\!\nmid q\,\ $ for all naturals $\,n > 1$
$(2)\qquad\quad p^2\!\nmid q\ \ $ for all primes $\,p$
$(3)\qquad\quad q\,$ is a product of distinct primes
$(4)\qquad\quad q\mid n^k\Rightarrow\, q\mid n\,\ $ for all naturals $\,n,k$
Proof $\ $ All $\,(n)\Rightarrow (n\!+\!1)$ are obvious. $\,(4)\,\Rightarrow\,(1)\,$ may be proved as follows
$\qquad\ \ $ If $\ q = an^2\,$ then $\ q\mid (an)^2\,\overset{ (4)}\Rightarrow\ q\mid an\,\Rightarrow\,an^2\mid n\,\Rightarrow\,n=1$
Remark $\ $ For further characterizations see here, which includes the little known $\, q^q\!\mid n^n\Rightarrow\, q\mid n$