1. Miniconference on the Mathematics of Computation
MTH 210
Modular Arithmetic 1
Dr. Anthony Bonato
Ryerson University

2. Even vs odd simple way to “split up” integers:
a number n is even if it has 2 as a divisor
we write 2 | n
otherwise, it is odd

3. use [0] to denote even integers
use [1] to denote odd integers
[0] = 0, ±2, ±4, ±6 …
[1] = ±1, ±3, ±5 …

4. Weird arithmetic [0] + [0] = [0] [0] + [1] = [1] [1] + [1] = [0]
“even plus even is even”
[0] + [1] = [1]
“even plus odd is odd”
[1] + [1] = [0]
“odd plus odd is even”

5. Weird arithmetic [0] x [0] = [0] [0] x [1] = [1] [1] x [1] = [1]
“even times even is even”
[0] x [1] = [1]
“even times odd is even”
[1] x [1] = [1]
“odd times odd is odd”

6. Congruences x ≡ y (mod 2) if 2 divides x – y write: 2 | (x-y)
eg: 4 ≡ 0 (mod 2), 17 ≡ 1 (mod 2)

7. Key facts x ≡ y (mod 2) then y ≡ x (mod 2) x ≡ x (mod 2)
If x ≡ y (mod 2) and y ≡ z (mod 2),
then x ≡ z (mod 2)

8. General congruences m > 2 an integer x ≡ y (mod n) if n | (x-y)
eg 13 ≡ 1 (mod 12), 23 ≡ 3 (mod 5)

9. Key facts x ≡ y (mod n) then y ≡ x (mod n) x ≡ x (mod n)
If x ≡ y (mod n) and y ≡ z (mod n),
then x ≡ z (mod n)

10. [2] = {x: remainder of 2 when 3 divides x}
[m] = {x | x ≡ m (mod n)}
called equivalence classes (mod n) or congruence classes
eg if n = 3, then
[2] = {x: remainder of 2 when 3 divides x}

11. Exercises

12. Miniconference on the Mathematics of Computation
MTH 210
Modular Arithmetic II
Dr. Anthony Bonato
Ryerson University

13. Key facts Given a ≡ c (mod n) and b ≡ d (mod n) a + b ≡ c + d (mod n)
For all integers k > 0,
ak ≡ ck (mod n)

14. Exercises