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E-STAR - Student

Home Mathematics MATHEMATICS: A Textbook for Grade 11   Chapter 1: Modular Arithmetic

E-Lecture - Important Points
  • A number which corresponds to the number of points on a regular polygon and discrete geometric pattern of equally spaced points is called a figurate number.
  • Two integers a and b are congruent modulo m if they differ by an integer multiple of m, that is, b a km for some k ∊ ℤ. This equivalence is written a ≡ b (mod m). We read “a ≡ b (mod m)” as “a is congruent to b modulo m”, which means m|a - b.
  • Let n be a positive integer. For all a, b ∊ ℤ
    • a ≡ a (mod m)
    • If a ≡ b (mod m), then b ≡ a (mod m).
    • If a ≡ b (mod m) and b ≡ c (mod m), then a ≡ c (mod m).
  • The sum of two or more numbers in a given modulo is found by first adding the numbers before converting to the given modulo. That is, if a ≡ b (mod m) and c ≡ d (mod m), then a + c = b + d (mod m).
  • The difference of two numbers in a given modulo is found by first subtracting the numbers before converting to the given modulo. That is, if a ≡ b (mod m) and c ≡ d (mod m), then a - c = b - d (mod m).
  • The product of two or more numbers in a given modulo is found by first multiplying the numbers before converting to the given modulo. That is, if a ≡ b (mod m) and c ≡ d (mod m), then
    a × c = b × d (mod m).
  • If the value of A is increased by a multiple of B (say, kB, where k is any integer), then
    A mod B = (A + kB) mod B; where k is any integer.
  • Addition property of modulo arithmetic is if A, B and C are positive integers, then
    (A + B) mod C = (A mod C + B mod C) mod C
  • Subtraction property of modulo arithmetic is if A, B and C are positive integers, then
    (A - B) mod C = (A mod C - B mod C) mod C
  • Multiplication property of modulo arithmetic is if A, B and C are positive integers, then
    (A × B) mod C = (A mod C × B mod C) mod C
  • Let a be any integer and d be a positive integer. Then, there exists unique q and
    r (0 ≤ r < d) such that
    a = dq + r.
    d is called the divisor, a is called the dividend, q is called the quotient and r is called the remainder.
  • Let a be any integer and d be a positive integer. If a = dq + r, then a mod d = r.
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