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We construct two non-isomorphic nuclear, stably finite, real rank zero 
${{C}^{*}}$-algebras 
$E$ and 
${{E}^{'}}$ for which there is an isomorphism of ordered groups 
$\Theta :{{\oplus }_{n\ge 0}}{{K}_{\bullet }}(E;\mathbb{Z}/n)\to {{\oplus }_{n\ge 0}}{{K}_{\bullet }}({E}';\mathbb{Z}/n)$ which is compatible with all the coefficient transformations. The ${{C}^{*}}$-algebras $E$ and ${{E}^{'}}$ are not isomorphic since there is no 
$\Theta$ as above which is also compatible with the Bockstein operations. By tensoring with Cuntz’s algebra 
${{\mathcal{O}}_{\infty }}$ one obtains a pair of non-isomorphic, real rank zero, purely infinite ${{C}^{*}}$-algebras with similar properties.
