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Proposition: Floor Function and Division with Quotient and Remainder

For two integers $a,b\in\mathbb Z$ with $a > 0,$ the quotient $q$ in the division with quotient and remainder $$b=qa+r,\quad 0\le r< a$$ can be exactly determined by floor function $q=\lfloor \frac ba\rfloor$, i.e. we have $$b=\left\lfloor\frac ba\right\rfloor a+r,\quad 0\le r< a.$$

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References

Bibliography

1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
2. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927