# Linear Congruence Relation and Complete Residue Systems

Xiquan Liang; Li Yan; Junjie Zhao

Formalized Mathematics (2007)

- Volume: 15, Issue: 4, page 181-187
- ISSN: 1426-2630

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topXiquan Liang, Li Yan, and Junjie Zhao. "Linear Congruence Relation and Complete Residue Systems." Formalized Mathematics 15.4 (2007): 181-187. <http://eudml.org/doc/267335>.

@article{XiquanLiang2007,

abstract = {In this paper, we defined the congruence relation and proved its fundamental properties on the base of some useful theorems. Then we proved the existence of solution and the number of incongruent solution to a linear congruence and the linear congruent equation class, in particular, we proved the Chinese Remainder Theorem. Finally, we defined the complete residue system and proved its fundamental properties.},

author = {Xiquan Liang, Li Yan, Junjie Zhao},

journal = {Formalized Mathematics},

language = {eng},

number = {4},

pages = {181-187},

title = {Linear Congruence Relation and Complete Residue Systems},

url = {http://eudml.org/doc/267335},

volume = {15},

year = {2007},

}

TY - JOUR

AU - Xiquan Liang

AU - Li Yan

AU - Junjie Zhao

TI - Linear Congruence Relation and Complete Residue Systems

JO - Formalized Mathematics

PY - 2007

VL - 15

IS - 4

SP - 181

EP - 187

AB - In this paper, we defined the congruence relation and proved its fundamental properties on the base of some useful theorems. Then we proved the existence of solution and the number of incongruent solution to a linear congruence and the linear congruent equation class, in particular, we proved the Chinese Remainder Theorem. Finally, we defined the complete residue system and proved its fundamental properties.

LA - eng

UR - http://eudml.org/doc/267335

ER -

## References

top- [5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
- [6] Józef Białas. Some properties of the intervals. Formalized Mathematics, 5(1):21-26, 1996.
- [7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
- [8] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
- [9] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
- [10] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
- [11] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.
- [12] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics, 9(1):191-196, 2001.
- [13] Yoshinori Fujisawa and Yasushi Fuwa. The Euler's function. Formalized Mathematics, 6(4):549-551, 1997.
- [14] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
- [15] Andrzej Kondracki. The Chinese Remainder Theorem. Formalized Mathematics, 6(4):573-577, 1997.
- [16] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.
- [17] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.
- [18] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83-86, 1993.
- [19] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.
- [20] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.
- [21] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.
- [22] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.
- [1] Grzegorz Bancerek. Arithmetic of non-negative rational numbers. To appear in Formalized Mathematics. Zbl06213832
- [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
- [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [4] Grzegorz Bancerek. Joining of decorated trees. Formalized Mathematics, 4(1):77-82, 1993.
- [23] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.
- [24] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
- [25] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [26] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
- [27] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
- [28] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990.