Division &amp; Divisibility

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# Division &amp; Divisibility - PowerPoint PPT Presentation

Division &amp; 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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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
• b+c = a.x + a.y
• = a(x + y)
• and that is divisible by a

Division

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

Division

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

Division

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

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

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?