Let a, b be positive integers and let x = gcd(a,b) and y=lcm(a,b) Show that $\displaystyle Z_{a} \oplus Z_{b}$ is isomorphic to $\displaystyle Z_{x} \oplus Z_{y}$.
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Originally Posted by aliceinwonderland Let a, b be positive integers and let x = gcd(a,b) and y=lcm(a,b) Show that $\displaystyle Z_{a} \oplus Z_{b}$ is isomorphic to $\displaystyle Z_{x} \oplus Z_{y}$. Use prime power decompositions. If $\displaystyle N = \prod_{j}p_j^{a_j}$ then $\displaystyle \mathbb{Z}_N \simeq \bigoplus_j \mathbb{Z}_{p_j^{a_j}}$
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