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Division & DivisibilityPowerPoint Presentation

Division & Divisibility

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## PowerPoint Slideshow about ' Division & Divisibility' - enrico

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- 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

- 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

- a|b a.m = b
- b.c = a.m.c
- which is divisible by a

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

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

The Division Algorithm (aint no algorithm)

dividend

divisor

remainder

quotient

- 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

NOTE: remainder r is positive and divisor d is positive

- 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

- 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?

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