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Proposition: Connection between Quotient, Remainder, Modulo and Floor Function

Let \(a,b\) be given natural numbers with \(a > 0\). The uniquely defined quotient and remainder $q,r$ in the equation \[b=qa+r,\quad 0\le r < a\] are given by $q=\lfloor \frac ba\rfloor$ (floor function) and $r\equiv b\mod a$ (congruence class $b$ modulo $a$).

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References

Bibliography

  1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition