1. Division & Divisibility

2. a divides b if a is not zero there is a m such that a.m = b a is a factor of b b is a multiple of a a|b Division

3. If a|b and a|c then a|(b+c) If a divides b and a divides c then a divides b plus c a|b a.x = b a|c a.y = c b+c = a.x + a.y = a(x + y) and that is divisible by a Division

4. a|b a.m = b b.c = a.m.c which is divisible by a Division

5. a|b a.x = b b|c b.y = c c = a.x.y and that is divisible by a Division

6. Theorem 1 (page 202, 6 th ed, page 154, 5 th ed)

7. The Division Algorithm (aint no algorithm) a is an integer and d is a positive integer there exists unique integers q and r, 0 r d a = d.q. + r a divided by d = q remainder r dividend divisor quotient remainder NOTE: remainder r is positive and divisor d is positive

8. Division a = d.q + r and 0 <= r < d a = -11 and d = 3 and 0 <= r < 3 -11 = 3q + r q = -4 and r = 1 a = d.q + r and 0 <= r < d a = -63 and d = 20 and 0 <= r <= 20 -63 = 20q + r q = -4 and r = 17 a = d.q + r and 0 <= r < d a = -25 and d = 15 and 0 <= r < 15 -25 = 15.q + r q = -2 and r = 10

9. Division a = d.q + r and 0 <= r < d a = -11 and d = 3 and 0 <= r < 3 -11 = 3q + r q = -4 and r = 1 Troubled by this? Did you expect q = -3 and r = -2? What if 3 of you went to a café and got a bill for £11? Would you each put £3 down and then leg it? Or £4 each and leave £1 tip?