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Properties of Integers

Objectives

At the end of this unit, students should be able to: State the division algorithm Apply the division algorithm Find the gcd using Euclidean algorithm Use the gcd to represent a linear combination of

two integers Find the least common multiple Represent an integer as a product of primes

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Contents

Division algorithm Greatest common divisor Fundamental Theorem of Arithmetic

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Division Algorithm

Definition 4.1

Given a, b Z and b 0,

b divides a, written as b | a,

if there is an integer n such that a = bn.

b is a divisor of a,

or a is a multiple of b.

E.g 2 | 8, 3 | 15, etc.

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Theorem 4.3

For all a, b, c Z (set of integers),

a. 1|a and a|0.

b. [(a | b) (b | a)] a = b.

c. [(a | b) (b | c)] a | c.

d. a | b a | bx for all x Z.

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Theorem 4.3 (cont.)

e. If x = y + z, for some x, y, z Z, and a divides two of the three integers x, y, and z, then a divides the remaining integer.

f. [(a | b) (a | c)] a | (bx + cy), for all x, y Z. (The expression bx + cy is called a linear combination of b, c.)

g. For 1 i n, let ci Z. If a divides each ci,

then a| (c1x1 + c2x2 + … + cnxn), where xi Z

for all 1 i n.

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Proof (part f)

If a|b and a|c, then b = am and c = an, for some m, n Z.

So bx + cy = (am)x + (an)y = a(mx + ny)

Since bx + cy = a(mx + ny), with mx + ny Z, it follows that

a | (bx + cy).

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Question

???

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Prime and Composite Numbers

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Prime Numbers

LEMMA 4.1

If n Z+ and n is composite, then there is a prime p such that p | n.

E.g 5 | 20 (5 is a prime and 20 a composite)

Theorem 4.4

There are infinitely many primes.

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Division Algorithm

Theorem 4.5

The Division Algorithm. If a, b Z, with b > 0, then there exist unique q, r Z

with a = qb + r, 0 r < b.

q: quotient

r: remainder

b: divisor

a: dividend.

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Greatest Common Divisor

Objectives:

On completion this unit, you should be able to:

1. Find the gcd using Euclidean algorithm

2. Use the gcd to represent a linear combination of two integers

3. Find the least common multiple

4. Represent an integer as a product of primes

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Contents

Greatest common divisor Least common multiple Fundamental Theorem of Arithmetic

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Common Divisor

c is a common divisor of a and b thenc | b and c | be.g. 3 | 6 and 3 | 9 3 is a common divisor of 6 and 9.1 | 42 and 1 | 702 | 42 and 2 | 707 | 42 and 7 | 7014 | 42 and 14 | 70(1,2, 7, 14) are common divisors.The greatest common divisor (gcd) is 14.

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Greatest Common Divisor

Definition 4.3

Let a, b Z, where either a 0 or b 0. Then c Z+ is called a greatest common divisor of a, b if

i) c|a and c|b (that is, c is a common divisor of a, b), and

ii) for any common divisor d of a and b, we have d|c. (i.e. d ≤ c)

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Greatest Common Divisor

Theorem 4.6For all a, b Z+, there exists a unique c Z+ that is the greatest

common divisor of a, b.

gcd(a, b) = gcd(b, a)For each a Z, if a 0, then

gcd(a, 0) = |a|.when a, b Z+, we have

gcd(–a, b) = gcd(a, –b) = gcd(– a, – b) = gcd(a, b)gcd(0, 0) not defined and is of no interest to us.

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Relatively Prime integers

From Theorem 4.6, not only does gcd(a, b) exist but that gcd(a, b) is also the smallest positive integer we can write as a linear combination of a and b.

Integers a and b are called relatively prime when gcd(a, b) = 1 that is, when there exist x, y Z with ax + by = 1.

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Question

???

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Activity

Find gcd(72,63), gcd(330,156)

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Least Common Multiple

Definition 4.4

For a, b, c Z+, c is called a common multiple of a, b if c is a multiple of both a and b.

Furthermore, c is the least common multiple of a, b if it is the smallest of all positive integers.

We denote c by lcm(a, b).

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Fundamental Theorem of Arithmetic

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