= Solution
Suppose
$$
n=p_1\cdots p_r=q_1\cdots q_s
$$
are two factorizations into primes. Repeatedly apply the cyclic-<subgroup> construction of part (c), first removing $p_r$, then $p_{r-1}$, and so on. This constructs a <composition series> for $C_n$ whose <composition factor>[composition factors] are $C_{p_1},\ldots,C_{p_r}$. The second factorization constructs another <composition series> with <composition factor>[composition factors] $C_{q_1},\ldots,C_{q_s}$.
By the <Jordan–Hölder theorem>, $r=s$ and these <composition factor>[composition factors] agree up to <group isomorphism> and permutation. Since isomorphic <finite group>[finite groups] have the same order, the corresponding primes agree. This proves \b[uniqueness of <prime factorization> up to permutation], using existence of prime factorizations but not their uniqueness.
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