Prime FactorsSlideshow 5, Mr Richard Sasaki, Room 307

Objectives

• Recall the meaning and list of prime numbers

• Understand how to calculate the product of prime factors for a number

• Use prime factors to show whether a rooted number produces an integer.

Prime NumbersWhat are prime numbers?Prime numbers are numbers with only two factors, the number itself and 1.

It’s useful to remember the first few prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …Obviously the list is infinite, but you should

know the first ones.If we divide a number by numbers in this list, we can find its prime factors.

Prime FactorsAn easy way to separate a number into a product of its prime factors is to create a prime factor tree.We try to divide the number by each of the prime numbers in the list and shrink it until it is only made of prime numbers.

602, 3, 5, 7, 11, …

② 30

② 15

③ ⑤22×3×5=60

Prime FactorsLet’s try with a larger number.

1960

2, 3, 5, 7, 11, …② 980

② 490

② 245

⑤ 49

⑦ ⑦

23×5×72=1960Try the worksheet!

Answers – Questions 1 - 6 12

② 6

② ③

22×3=12

30

② 15

③ ⑤2×3×5=30

50

② 25

⑤ ⑤2×52=50

42

② 21

③ ⑦

2×3×7=4 2

75

③ 25

⑤ ⑤

3×52=75

36

② 18

② 9

22×32=36

③③

Answers – Questions 7 - 12150

② 75

③ 25

2×3×52=150⑤⑤

770

② 385

⑤ 77

2×5×7×11=770⑪⑦

85

⑤ ⑰5×17=85

4620

② 2310

② 1155

③ 385⑤ 77

⑦ ⑪22×3×5×7×11=4620

189

③ 63

③ 21

⑦③

33×7=189

1001

⑦ 143

⑪ ⑬

7×11×13=1001

Writing in the form If we can write a number in the form where , then .Example

Show that produces an integer.

As 16 = 2, must be an integer ( =). 4 4

We can also do this by using the number’s prime factors.

Writing in the form ExampleWrite 81 as a product of its prime factors and hence, show that 81 is a square number.

81

③ 27

③ 9

③ ③

34=81()2=813×3

As we expressed 81 in the form , it must be square.

Writing in the form Example

Write 132 as a product of its prime factors and show that is not an integer.

132

② 66

② 33

③ ⑪

22×3×11=132

We cannot write in the form so 132 is not an integer.

Answers – Question 19

③ ③

32=9

100

② 50② 25

⑤ ⑤

22×52=102

132

② 66② 33

③ ⑪

22×3×11≠𝑏2256

② 128② 64

② 32

② 16

② 8② 4②

28=162

400② 200② 100② 50

② 25

⑤ ⑤

24×52=202

142

② 71

2×71≠𝑏2

②

Answers – Question 2225

③ 75③ 25⑤ ⑤

32×52=152

999③ 333③ 111③ 37

33×37≠𝑏2

289

⑰ ⑰

172=289260

② 130② 65

⑤ ⑬

22×5×13≠𝑏2

784② 392② 196② 98② 49⑦ ⑦24×72=282

6258

②3129

③1043

⑦ 149

2×3×7×149≠𝑏2