Mod arithmetic.

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Presentation transcript:

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 a is congruent to b modulo m if m divides a - b

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. 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 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 proof

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)