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Given three positive integers $a,b,c$, a proportionally modular Diophantineinequality is an expression of the form $ax \mod{b} \le cx$. Our aim is to givea recursive formula for the least solution to such an inequality. We then usethe formula to derive an algorithm. Finally, we apply our results to a questionof Rosales and Garc\'ia-S\'anchez.

On the least positive solution to a proportionally modular Diophantine inequality