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  • https://math.libretexts.org/Courses/Las_Positas_College/Math_for_Liberal_Arts/05%3A_Numeration_Systems/5.04%3A_Modular_Systems
    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 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_Arithmetic
    The notation “(mod n)” after m1m2 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 m1m2 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, mmodn{0,1,2,,n1}. 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_Arithmetic
    The 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_Relations
    This 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/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/07%3A_Equivalence_Relations/7.04%3A_Modular_Arithmetic
    The 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/Courses/Mount_Royal_University/Higher_Arithmetic/3%3A_Modular_Arithmetic/3.2%3A_Modulo_Arithmetic
    Let a,b,c,d,Z such that ab(modn) and cd(modn). Then (a+c)(b+d)(modn). Let a,b,c,dZ, such that \(a \equiv b (mod...Let a,b,c,d,Z such that ab(modn) and cd(modn). Then (a+c)(b+d)(modn). Let a,b,c,dZ, such that ab(modn) and cd(modn). Let a,b,c,d,Z such that ab(modn) and cd(modn). Then (ac)(bd)(modn). Let a,b,c,dZ, such that ab(modn) and cd(modn).
  • https://math.libretexts.org/Courses/Florida_SouthWestern_State_College/MGF_1131%3A_Mathematics_in_Context__(FSW)/01%3A__Number_Representation_in_Different_Bases_and_Cryptography/1.06%3A__Modular_Arithmetic
    This section explores modular arithmetic, or clock arithmetic, emphasizing its practical applications in scenarios like time calculations and scheduling. It explains the concept of modulus, computatio...This section explores modular arithmetic, or clock arithmetic, emphasizing its practical applications in scenarios like time calculations and scheduling. It explains the concept of modulus, computation with negative numbers, and includes examples and exercises, particularly focusing on modulo 24 for hours, modulo 7 for days, and modulo 12 for months.

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