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Division & Divisibility

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Division & Divisibility. Division. 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. 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

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Presentation Transcript
slide2

Division

  • 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
slide3

Division

  • 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
slide4

Division

  • a|b  a.m = b
  • b.c = a.m.c
  • which is divisible by a
slide5

Division

  • a|b  a.x = b
  • b|c  b.y = c
  • c = a.x.y
  • and that is divisible by a
slide6

Division

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

slide7

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

slide8

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
slide9

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?

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