同餘(congruence) 的概念就是將整數適當的分成有限多類,使其仍能和整數一樣的運算, 從而得到一些整數的重要性質. 本章就是探討congruence的定義以及得到一些有關congruence 的重要式子.

BASIC PROPERTIES OF CONGRUENCES The letters a; b; c; d; k represent integers. The letters m; n represent positive integers. The notation a b (mod m) means that m ...

Cor: If a ≡ a’ (mod n), then replacing a by a’ in any arithmetic formula gives an ≡ (mod n) formula congruence.18 Congruence mod n Corollary: If a ≡ b (mod n) & cc ≡ d (mod n), then a+c ⋅ ≡ …

In congruence modulo 2 we have [0]2 = f0; 2; 4; 6; g [1]2 = f 1; 3; 5; 7; g : Thus, the congruence classes of 0 and 1 are, respectively, the sets of even and odd integers.

We will denote the collection of congruence classes by Z/nZ: Z/nZ = {a + nZ | a ∈ Z}. Before we give more examples, it will be convenient to give a complete description of Z/nZ.

First, we solve the congruence modulo 3: testing all 3 possible residues shows that the only solution is x 1 (mod 3). Now we just compute the derivative: if q(x) = x3 2x + 7, then q0(x) = …

Notice that under congruence mod 5 all of the integers group into collections which are congruent to one another mod 5: ... Every integer is in one of these lines. It follows that we can divide all …

Congruence Definitions Let m ∈ N. • For any a, b ∈ Z, we say “a is congruent to b modulo m” if m | (b − a). In symbols, we write this as a ≡ b mod m. • We can use congruence to define a …

Therefore, up to the use of additive notation, we can generalize congruence modulo n in Z to an arbitrary group G as follows. First we replace nZ by a subgroup H < G.

The meaning of CONGRUENCE is the quality or state of agreeing, coinciding, or being congruent. How to use congruence in a sentence.