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

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

7. 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

11. 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

14. 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

16. 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( )
h( )
h( )
h( )