16. (I) Let $n=c_{k} \cdot 10^{k}+\cdots+c_{1} \cdot 10+c_{0}$. Prove:
(i) $2|n \Longleftrightarrow 2| c_{0}$;
(ii) $5|n \Longleftrightarrow 5| c_{0}$;
(iii) $3|n \Longleftrightarrow 3|\left(c_{k}+\cdots+c_{0}\right)$;
(iv) $9|n \Longleftrightarrow 9|\left(c_{k}+\cdots+c_{0}\right)$;
(v) $11|n \Longleftrightarrow 11|\left(c_{k}-c_{k-1}+\cdots+(-1)^{k} \cdot c_{0}\right)$.
(II) Let $n=c_{k} \cdot(100)^{k}+\cdots+c_{1} \cdot(100)+c_{0}$. Prove:
(i) $11|n \Longleftrightarrow 11|\left(c_{k}+\cdots+c_{0}\right)$;
(ii) $101|n \Longleftrightarrow n|\left(c_{k}-c_{k-1}+\cdots+(-1)^{k} c_{0}\right)$.
(III) Let $n=c_{k} \cdot(1000)^{k}+\cdots+c_{1} \cdot(1000)+c_{0}$. Prove:
(i) $37|n \Longleftrightarrow 37|\left(c_{k}+\cdots+c_{0}\right)$;
(ii) $7|n \Longleftrightarrow 7|\left(c_{k}-c_{k-1}+\cdots+(-1)^{k} \cdot c_{0}\right)$;
(iii) $13|n \Longleftrightarrow 13|\left(c_{k}-c_{k-1}+\cdots+(-1)^{k} c_{0}\right)$.
(IV) Use the above results to propose corresponding divisibility tests for integers by $2,3,5,7,9,11,13,37$, or 101. Use this method to factorize the numbers 1535625, 1158066, 82798848, and 81057226635000 into their prime factors.