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- https://math.libretexts.org/Courses/Las_Positas_College/Math_for_Liberal_Arts/05%3A_Numeration_Systems/5.04%3A_Modular_SystemsIn terms of division, we say that \(a\) divides \(b\) if and only if the remainder is zero when \(b\) is divided by \(a\). In the late third century, the Chinese mathematician Sun Tzu asked his studen...In terms of division, we say that \(a\) divides \(b\) if and only if the remainder is zero when \(b\) is divided by \(a\). In the late third century, the Chinese mathematician Sun Tzu asked his students: "We have things of which we do not know the number; if we count by threes, the remainder is 2; if we count be fives, the remainder is 3; if we count by sevens, the remainder is 2.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/05%3A_Basic_Number_Theory/5.07%3A_Modular_ArithmeticThe notation “(mod \(n\))” after \(m_1\equiv m_2\) indicates a congruence relation, in which “mod \(n\)” are enclosed by a pair of parentheses, and the notation is placed at the end of the congruence....The notation “(mod \(n\))” after \(m_1\equiv m_2\) indicates a congruence relation, in which “mod \(n\)” are enclosed by a pair of parentheses, and the notation is placed at the end of the congruence. Given any integer \(m\), \[m\bmod n \in\{0,1,2,\ldots,n-1\}. \nonumber\] We call these values the residues modulo . In modular arithmetic, when we say “reduced modulo ,” we mean whatever result we obtain, we divide it by \(n\), and report only the smallest possible nonnegative residue.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/07%3A_Equivalence_Relations/7.04%3A_Modular_ArithmeticThe term modular arithmetic is used to refer to the operations of addition and multiplication of congruence classes in the integers modulo n.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/07%3A_Equivalence_Relations/7.04%3A_Modular_ArithmeticThe term modular arithmetic is used to refer to the operations of addition and multiplication of congruence classes in the integers modulo n.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/04%3A_Relations/4.03%3A_Equivalence_RelationsThis page explores equivalence relations in mathematics, detailing properties like reflexivity, symmetry, and transitivity. It defines equivalence classes and provides checkpoints for assessing equiva...This page explores equivalence relations in mathematics, detailing properties like reflexivity, symmetry, and transitivity. It defines equivalence classes and provides checkpoints for assessing equivalence in subsets, modular arithmetic, and integer divisibility.
- https://math.libretexts.org/Courses/Mount_Royal_University/Higher_Arithmetic/3%3A_Modular_Arithmetic/3.2%3A_Modulo_ArithmeticLet \( a, b, c,d, \in \mathbb{Z}\) such that \(a \equiv b (mod\,n) \) and \(c \equiv d (mod\, n). \) Then \((a+c) \equiv (b+d)(mod\, n).\) Let \(a, b, c, d \in\mathbb{Z}\), such that \(a \equiv b (mod...Let \( a, b, c,d, \in \mathbb{Z}\) such that \(a \equiv b (mod\,n) \) and \(c \equiv d (mod\, n). \) Then \((a+c) \equiv (b+d)(mod\, n).\) Let \(a, b, c, d \in\mathbb{Z}\), such that \(a \equiv b (mod\, n) \) and \(c \equiv d (mod \,n). \) Let \( a, b, c,d, \in \mathbb{Z}\) such that \(a \equiv b (mod \, n) \) and \(c \equiv d (mod \,n). \) Then \((ac) \equiv (bd) (mod \,n).\) Let \(a, b, c, d \in \mathbb{Z}\), such that \(a \equiv b (mod\, n) \) and \(c \equiv d (mod \, n). \)
