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- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/03%3A_Constructing_and_Writing_Proofs_in_Mathematics/3.01%3A_Direct_ProofsIf a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For each integer a, if there exists an integer \(...If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For each integer a, if there exists an integer n such that a divides (8n+7) and a divides (4n+1), then a divides 5.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/03%3A_Constructing_and_Writing_Proofs_in_Mathematics/3.05%3A_The_Division_Algorithm_and_CongruenceRecall that if a and b are integers, then we say that a is congruent to b modulo n provided that n divides a−b, and we write a≡b (mod n). (See Section 3.1....Recall that if a and b are integers, then we say that a is congruent to b modulo n provided that n divides a−b, and we write a≡b (mod n). (See Section 3.1.) We are now going to prove some properties of congruence that are direct consequences of the definition.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji)/03%3A_CongruencesA congruence is nothing more than a statement about divisibility. The theory of congruences was introduced by Carl Friedreich Gauss. Gauss contributed to the basic ideas of congruences and proved seve...A congruence is nothing more than a statement about divisibility. The theory of congruences was introduced by Carl Friedreich Gauss. Gauss contributed to the basic ideas of congruences and proved several theorems related to this theory. We start by introducing congruences and their properties. We proceed to prove theorems about the residue system in connection with the Euler ϕ -function.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/03%3A_Constructing_and_Writing_Proofs_in_Mathematics/3.01%3A_Direct_ProofsIf a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For each integer a, if there exists an integer \(...If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false. (a) For each integer a, if there exists an integer n such that a divides (8n+7) and a divides (4n+1), then a divides 5.
- https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_II_(Illustrative_Mathematics_-_Grade_8)/01%3A_Rigid_Transformations_and_Congruence/1.03%3A_Congruence
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/03%3A_Constructing_and_Writing_Proofs_in_Mathematics/3.05%3A_The_Division_Algorithm_and_CongruenceRecall that if a and b are integers, then we say that a is congruent to b modulo n provided that n divides a−b, and we write a≡b (mod n). (See Section 3.1....Recall that if a and b are integers, then we say that a is congruent to b modulo n provided that n divides a−b, and we write a≡b (mod n). (See Section 3.1.) We are now going to prove some properties of congruence that are direct consequences of the definition.
