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Remainder classes Every positive integer m allows to define in the set Z of integers a relation of the following kind - do my html homework : If a and b are integers, then a is called congruent b modulo m if their difference a-b is an integer multiple q⋅m of m.

In characters: a≡b mod m or a≡b(m) for short. Examples:

8≡3(5) means 8-3=1⋅5, and -7≡8(5) means -7-8=-15=(- 3)⋅5.

But it is not 11≡2(5), since 11-2=9≠g⋅5.

It is significant that for every integer m>0, an equivalence relation in Z is given by the relation a≡b mod m, which divides Z into equivalence classes.

Theorem: Let a, b and m be integers with m>0.

The relation a≡b(m) is an equivalence relation in Z, called congruence modulo m. Proof:

(1) The congruence modulo m is reflexive, since for all a from Z holds:  a≡a(m) (because a-a=0⋅m).

(2) The congruence modulo m is symmetric, since for all a, b from Z holds:

If a≡b(m), then also b≡a(m), because a≡b(m) is equivalent to a-b=g⋅m. Thus -(a-b)=(-g)⋅m, i.e. b-a=(-g)⋅m.

(3) The relation is also transitive - geometry homework helper , i.e. for all a, b, c from Z it follows from a≡b(m) and b≡c(m) also a≡c(m), because a≡b(m) and b≡c(m) is equivalent to the equations a-b=g1⋅m and b-c=g2⋅m. The result follows from adding the two equations: a-c=(a-b)+(b-c)=g1⋅m+g2⋅m=(g1+g2)⋅m=g⋅m.

Definition: The classes of the class division in Z, which corresponds to the congruence modulo m, are called residue classes modulo m - do my homework . The notation [a] m denotes that residue class which contains the number a.

For m=5 one obtains the following residue classes:

 5={..., -10, -5, 0, 5, 10, ...}

[ 1 ] 5={..., -9, -4, 1, 6, 11, ...}

 5={..., -8, -3, 2, 7, 12, ...}

 5={..., -7, -2, 3, 8, 13, ...}

 5={..., -6, -1, 4, 9, 14, ...}

The name residue class is explained by the following context:

Theorem: It holds a≡b(m) exactly if a and b, when divided by m, leave the same remainder r with 0≤r<m.

It follows that there are exactly m remainder classes modulo m, namely  m, [ 1 ] m, ..., [m-1] m. The numbers 0, 1, ..., m-1 form a complete representative system of the classes modulo m, which is called the smallest non-negative representative system.

Liberal theoryes

Definition according to Arrhenius

Marco Polo (1253-1324)

Cosquer Cave

The invention of Gutenberg

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