Prime and Composite Numbers

Greatest Common Divisor &Least Common Multiple

A Prime Number has exactly two distinct positive divisors.

A Prime Number can only be divided by itself and 1.Prime Numbers:

2357111317192329…

A composite number has factors other than itself and 1.

Example: 6 is composite because its factors are 1, 2, 3 and 6.

A factor tree can be used to find the prime factors of a composite number.

24

46

2 23 2

24

38

2

2

4

2Prime factorization

of 24:24 = 2 · 2 · 2 · 3 or 24 = 23 · 3

Fundamental Theorem of Arithmetic:Each composite number can be written as a product of primes in one way only.

Is 43 prime or composite?If 43 is composite, it has prime factors

Divide 43 by prime numbers to see if they are divisors43 2 = 43 3 = 43 5 = 43 7 = 43 11 =

43 is prime.

21 R 114 R 18 R 36 R 1 3 R 10 How far did we

really have to go before we could determine that 43 is prime?

What is the largest possible prime factor of 43?

Prime2357

111317

Square49

2549

121169289

43

5

43 2 = 43 3 = 43 5 = 43 7 = 43 11 =

21 R 114 R 18 R 36 R 1 3 R 10

Is 113 prime or composite?

Prime2357

111317

Square49

2549

121169289

113

113 is prime

Is 2 a divisor of 113?Is 3 a divisor of 113?Is 5 a divisor of 113?Is 7 a divisor of 113?

NoNoNoNo

Divisors of Composite Numbers

List all divisors of 12

How many divisors does 1500 have?

1, 2, 3, 4, 6, 12

1500

10015

3 105 10

2 5 2 5

1500 = 22 · 31 · 53

Number of divisors:(2 + 1)(1 + 1)(3 + 1)= (3)(2)(4)= 24

The Greatest Common Divisor (GCD) of two or more integers is the largest integer that divides the numbers.

Find the GCD of 8 and 20

Intersection of sets method

Factors of 8: Factors of 20:

Common Factors:

GCD(8, 20) =

1, 2, 4, 81, 2, 4, 5, 10, 20

1, 2, 4

4

Find the GCD of 8 and 20

Prime factorization method

8 = 2 · 2 · 220 = 2 · 2 · 5

GCD(8, 20) = 2 · 2 = 4

2 2

4

2 2

20

54

2 2

8

2

Find the GCD of 42 and 63

Intersection of sets method

Factors of 42: Factors of 63:

Common Factors:

GCD(42, 63) =

1, 2, 3, 6,1, 3, 7, 9, 21, 63

1, 3, 7,

21

7, 14, 21, 42,

21

Find the GCD of 42 and 63

Prime factorization method

42 = 2 · 3 · 7 63 = 3 · 3 · 7

GCD(42, 63) = 3 · 7 = 21

3 7

9

3 3

63

76

2 3

42

7

Euclidean AlgorithmThe GCD(a, b) = GCD(r, b) where r is the remainder when a is divided by b.

GCD(42, 63)

142 63 42 21

221 42 42 0 0 remainder

GCD = 21

GCD(1824, 7448)

41824 7448 7296 152

1152 1824 152 304 304

0 0 remainder

GCD = 152

2

Check: Is 152 a the GCD of 1824 and 7448?1824 ÷ 152 = 127448 ÷ 152 = 49

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the numbers.

Find LCM(8, 20)

Intersection of sets method:

Multiples of 8:Multiples of 20:

Common Multiples:

LCM:

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …20, 40, 60, 80, …

40, 80, 120, …

40

Find LCM(8, 20)

Prime factorization method

8 = 2 · 2 · 220 = 2 · 2 · 5

LCM(8, 20) = 2 · 2 · 2 · 5 = 40

2 2

4

2 2

20

54

2 2

8

2

2 5 L C MEFTOVERS

OUNT

with

ULTIPLES