1. Math 51/COEN 19

4. Number theory vocab Let a, b Z, a ≠ 0. We say that a divides b and write a|b if m Z such that b = ma (ie b is a multiple of a)

5. Number theory vocab Let a, b Z, a ≠ 0. We say that a divides b and write a|b if m Z such that b = ma (ie b is a multiple of a) Theorem: Let a, b, c, Z with a, b ≠ 0 – i. If a|b and a|c then a|(b+c) and a|(b-c) – ii. If a|b then a|bc – iii. If a|b and b|c then a|c – iv. If a|b and a|c then a|mb+nc whenever m, n Z

7. Number theory vocab Let a, b Z, a 0. We say that a divides b and write a|b if m Z such that b = ma (ie b is a multiple of a) Theorem: Let a, b, c, Z with a, b 0 – i. If a|b and a|c then a|(b+c) and a|(b-c) – ii. If a|b then a|bc – iii. If a|b and b|c then a|c – iv. If a|b and a|c then a|mb+nc whenever m, n Z Division algorithm: Let a Z and d Z +. Then there are unique integers q and r such that a = qd+r and 0 ≤ r<d

9. Number theory vocab Division algorithm: Let a Z and d Z +. Then there are unique integers q and r such that a = qd+r and 0 ≤ r<d Let m Z, m ≥ 2. If a, b Z and m|(a-b) we write (a ≡ b)mod m (ie a and b differ by a multiple of m)

11. Number theory vocab Division algorithm: Let a Z and d Z +. Then there are unique integers q and r such that a = qd+r and 0 ≤ r<d Let m Z, m ≥ 2. If a, b Z and m|(a-b) we write (a ≡ b)mod m (ie a and b differ by a multiple of m) Theorem: Let a, b, m Z, m ≥ 2. If (a ≡ b)mod m then i)(b ≡ a)mod m ii) (a ≡ a)mod m iii) if (a ≡ b)mod m and (b ≡ c)mod m then (a ≡ c)mod m (prove in HW)

12. Which are congruent to each other mod 5? -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

15. Theorem: if (a ≡ b)mod m and (c ≡ d)mod m then (a+c ≡ b+d)mod m and (a-c ≡ b-d)mod m and (ac ≡ bd)mod m (won’t prove)

17. Let n Z. i) n is even iff (n ≡ 0)mod 2 ii) n is odd iff (n ≡ 1)mod 2