I found this problem and I'm not sure if I can apply the reminder theorem on it. Are $\mathbb{Z}_8\oplus\mathbb{Z}_{18}$ and $\mathbb{Z}_2\oplus\mathbb{Z}_{72}$ isomorphic? I tend to believe so, but I cannot prove it. Can someone help, please?
2026-03-25 09:33:32.1774431212
Direct sum isomorphism question
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By the Chinese Remainder Theorem, $C_2\times C_{72}\cong C_2\times C_9\times C_8\cong C_{18}\times C_8$, since $\gcd(9,8)=\gcd(9,2)=1.$