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When solving $ax \equiv b \mod c$, is it okay to replace $a$ with any $m$ where $m \equiv a$?

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Yes, you can replace $a$ by any $a'$ such that $a'\equiv a\pmod{c}$. –  André Nicolas Jan 29 '13 at 23:34
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Note that "$m\equiv a$" is meaningless --- there must be a modulus. If you meant $m\equiv a\pmod c$, then you're OK. –  Gerry Myerson Jan 30 '13 at 0:09
    
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If you think about what $ax \equiv b \pmod c$ means, i.e. there exists a $ q \in \Bbb{Z}$ such that $ax = b + qc$ so if we have $a = m + qc$ Then substitution gives $$(m+qc)x = b + qc$$ and $$mx = b + (q-qx)c$$ So $$mx \equiv b \pmod c$$

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