The following proposition demonstrates the division with quotient and remainder we have introduced already provides an equivalent possibility to define congruences.
Proposition: Congruences and Division with Quotient and Remainder
a&=&q_am+r_a&0\le r_a < m,\\
b&=&q_bm+r_b&0\le r_b < m.
Then $a$ is congruent to $b$ if and only if they have the same remainder, formally
$$a\equiv b(m)\Longleftrightarrow r_a=r_b.$$
| | | | | created: 2019-04-10 21:28:19 | modified: 2019-04-10 21:54:59 | by: bookofproofs | references: , 
1.Proof: (related to "Congruences and Division with Quotient and Remainder")
This work is a derivative of:
Bibliography (further reading)
 Jones G., Jones M.: “Elementary Number Theory (Undergraduate Series)”, Springer, 1998
 Landau, Edmund: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927