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Proof

(related to Proposition: Connection between Quotient, Remainder, Modulo and Floor Function)

• Let \(a,b\) be given natural numbers with \(b\neq 0\).
• From definition of quotient and remainder we have that the equation \(b=qa+r\) uniquely determines two other natural numbers \(q,r\) with \(0\le r < a\).
• We have already shown that $q=\lfloor \frac ba\rfloor.$
• From congruences and division with quotient and remainder, it follows that \(r=b\mod a\).
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References

Bibliography

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