Proof
(related to Proposition: Counting the Solutions of Diophantine Equations of Congruences)
 By hypothesis, $m_1,\ldots,m_r$ are positive integers which are pairwise coprime.
 The Diophantine equation of congruences $f(x)(m_1\cdots m_r)\equiv 0(m_1\cdots m_r)\label{eq:E18602a}\tag{1}$ has only solutions if and only if the equations $f(x)(m_i)\equiv 0(m_i)\label{eq:E18602b}\tag{2}$ do for $i=1,\ldots,r.$
 Therefore, according to the Chinese remainder theorem, there is a bijection between the solutions of $(\ref{eq:E18602a})$ modulo the product $(m_1\cdots m_r)$ and the solutions of $(\ref{eq:E18602b})$ modulo the $r$ numbers $m_1,m_2,\ldots m_r.$
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References
Bibliography
 Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927