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mod arithmetic

mod arithmetic. mod arithmetic. a mod m is the remainder of a divided by m a mod m is the integer r such that a = qm + r and 0 <= r < m again, r is positive. Examples 17 mod 3 = 2 17 mod 12 = 5 (5 o’clock) -17 mod 3 = 1. congruences.

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mod arithmetic

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  1. mod arithmetic

  2. mod arithmetic • a mod m is the remainder of a divided by m • a mod m is the integer r such that • a = qm + r and 0 <= r < m • again, r is positive • Examples • 17 mod 3 = 2 • 17 mod 12 = 5 (5 o’clock) • -17 mod 3 = 1

  3. congruences a is congruent to b modulo m if m divides a - b

  4. proof a is congruent to b mod m if and only if the remainder of a divided by m is equal to the remainder of b divided by m.

  5. proof If a is congruent to b mod m and c is congruent to d mod m then a+c is congruent to b+d mod m

  6. proof If a is congruent to b mod m and c is congruent to d mod m then ac is congruent to bd mod m

  7. Mod arithmetic examples • -133 mod 9 = 2 (but in Claire?) • list 5 numbers that are congruent to 4 modulo 12 • hash function h(k) = k mod 101 • h(104578690) • h(432222187) • h(372201919) • h(501338753)

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