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Let $\\omega_{p_1p_2}:=(1+\\sqrt{p_1p_2})/2$ and let $\\Psi(\\omega_{p_1p_2})$ be the Hirzebruch sum of $\\omega_{p_1p_2}$. We show that $h(-p_1)h(-p_2)\\equiv h(p_1p_2)\\Psi(\\omega_{p_1p_2})/n$ $(\\text{mod }8)$, where $n=6$ (respectively, $n=2$) if $\\min\\{p_1,p_2\\}>3$ (respectively, otherwise). 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