Division and GCD

CSC2110 Tutorial 7

Darek Yung

Outline

Self Introduction Announcement Quick Review Example Q & A

Self Introduction

Yung Chun Kong, Darek Responsible for

• Topics in Number Theory • Tutorial 7 – 9• The third class work

Office: SHB 115 Office hour: Wed – Fri, 3pm – 7pm Feel free to raise question in newsgroup

Announcement

You can collect your class work 1 from Tom (SHB 115)

Quick Review

Divisibility Division Theorem GCD (Extended) Euclid’s GCD Algorithm Linear Combination Die Hard

Divisibility

If a | b, then a | bc for all c. If a | b and b | c, then a | c. If a | b and a | c, then a | sb + tc for all s and

t. For all c ≠ 0, a | b if and only if ca | cb.

Division Theorem

a = qb + r, 0 r < b q, r are unique

GCD

Every common divisor of a and b divides GCD(a, b)

GCD(ka, kb) = k * gcd(a,b) for all natural number k

If GCD(a, b) = GCD(a, c) = 1, GCD(a, bc) = 1

If a | bc and GCD(a, b) = 1, a | c GCD(a, b) = GCD(b, rem(a, b))

(Extended) Euclid’s GCD Algorithm

By Division Theorem• a = qb + r

And GCD property• GCD(a, b) = GCD(b, rem(a, b))

We will have examples later

Linear Combination

sa + tb is a linear combination of a and b Smallest positive linear combination, SPC SPC(a, b) = GCD(a, b)

Die Hard

Goal: want to z gallon of water by using jugs with x and y gallon capacities

Solved by SPC = GCD, i.e. EE GCD Algorithm

Example

True / False

• Q1: False, e.g. when x is multiple of p• Q2: False, e.g. when y is multiple of x• Q3: True, definition of prime

Example

True / False

• Q4: False, e.g. when s = t = 1• Q5: False, e.g. when m = 2, a = 4, s=t=b=1• Q6: False, the smallest POSITIVE one

Example

True / False

• Q7: True, consider: ka + kb = k(a+b)• Q8: False, should be “divides”

e.g. GCD(12,18) = 6, 2 | 6

• Q9: True, implies GCD(a, b) = GCD(a, c) = 1• Q10: True, GCD(a, bc) >= GCD(a, c)

Example

True / False

• Q11: False, when z is multiple of GCD(x, y)• Q12: False, e.g. consider z = x // z = GCD(x, y)

Example

True / False

• Q13: False, e.g. when z is not multiple of x, y = xz

• Q14: False, e.g. when y is multiple of x, z is not• Q15: True,

always able to find x, s.t. GCD(x,y)=1 when y is given

Example

Find GCD(564, 978)• 978 = 1(564) + 414• 564 = 1(414) + 150• 414 = 2(150) + 114• 150 = 1(114) + 36• 114 = 3(36) + 6• 36 = 6(6) + 0GCD(564, 978) = 6Euclid’s GCD AlgorithmCaution: No mark will be given if steps are skipped!!!(Marks for steps will be given even if answer is wrong)

Example

Find SPC(364, 516) and values of s, t that

364s + 516t = SPC(364, 516)

Example

Find SPC(364, 516)• SPC(364, 516) = GCD(364, 516)• 516 = 1(364) + 152• 364 = 2(152) + 60• 152 = 2(60) + 32• 60 = 1(32) + 28• 32 = 1(28) +4• 28 = 7(4) + 0SPC(364, 516) = GCD(364, 516) = 4How to find s and t???

Example

• 516 = 1(364) + 152 152 = 516 – 1(364)

• 364 = 2(152) + 60 60 = 364 – 2(152) = -2(516) + 3(364)

• 152 = 2(60) + 32 32 = 152 – 2(60) = 5(516) – 7(364)

• 60 = 1(32) + 28 28 = 60 – 1(32) = -7(516) + 10(364)

• 32 = 1(28) +4 4 = 32 – 1(28) = 12(516) – 17(364)

• 28 = 7(4) + 0 s = -17, t = 12 Extended Euclid’s GCD Algorithm

Example

Find SPC(152, 376) and values of s, t that

152s + 376t = SPC(152, 376)

Example

• SPC(152, 376) = GCD(152, 376)• 376 = 2(152) + 72

72 = 376 – 2(152)

• 152 = 2(72) + 8 8 = 152 – 2(72) = -2(376) + 5(152)

• 72 = 9(8) + 0 SPC(152, 376) = 8, with s = 5, t = -2

Example

Find GCD(42, 56, 98)• GCD(42, 56, 98) = GCD(42, GCD(56, 98))• GCD(56, 98) = 14 (by Euclid’s GCD Algorithm)• GCD(40, 14) = 2 (by Euclid’s GCD Algorithm)

GCD(42, 56, 98) = 2

Example

Show how to get 81 gallons of water by using 366 and 297 gallon jugs.• 366 = 1(297) + 69

69 = 366 – 1(297)• 297 = 4(69) + 21

21 = 297 – 4(69) = -4(366) + 5(297)• 69 = 3(21) + 6

6 = 69 – 3(21) = 13(366) – 16(297)• 21 = 3(6) + 3

3 = 21 – 3(6) = -43(366) + 53(297)• 6 = 2(3) + 0 -43(366) + 53(297) = 3 -1161(366) + 1431(297) = 81

Example

How to save water (reduce number of rounds of transferring water)?• -297(366) + 366(297) = 0• -891(366) + 1098(297) = 0

-(1161-891)(366) + (1431-1098)(297) = 81

-270(366) + 333(297) = 81

Q & A