1. Slideshow 5, Mr Richard Sasaki, Room 307
Prime Factors
Slideshow 5, Mr Richard Sasaki, Room 307

2. 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.

3. Prime Numbers What 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.

4. Prime Factors
An 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. 60
2, 3, 5, 7, 11, … 30 ②
2 2 ×3×5=60 15 ② ③ ⑤

5. Prime Factors Let’s try with a larger number. 1960 ② 980
2, 3, 5, 7, 11, … ②
490
2 3 ×5× 7 2 =1960 ②
245
Try the worksheet! ⑤ 49 ⑦ ⑦

6. Answers – Questions 1 - 6 12 30 50 ② 6 ② 15 ② 25 ② ③ ③ ⑤ ⑤ ⑤ 2 2 ×3=12
2×3×5=30
2× 5 2 =50 75 36 42 ③ 25 ② 18 ② 21 ⑤ ⑤ ② 9 ③ ⑦
3× 5 2 =75
2×3×7=42 ③ ③
2 2 × 3 2 =36

7. Answers – Questions 7 - 12 150 770 85 ② 75 ② 385 ⑤ ⑰ ③ 25 ⑤ 77 5×17=85⑦ ⑪
2× 3×5 2 =150
2×5×7×11=770
1001
4620
189 ⑦
143 ②
2310 ③ 63 ②
1155 ⑪ ⑬ ③
385 ③ 21
7×11×13 =1001 ⑤ 77 ③ ⑦ ⑦ ⑪
2 2 ×3×5×7×11=4620
3 3 ×7=189

8. Writing 𝑎 in the form 𝑏 2
If we can write a number 𝑎 in the form 𝑏 2 where 𝑏∈ℤ, then 𝑎 ∈ℤ.
Example
Show that produces an integer.
As 16 = 2, must be an integer ( 16 =± ).
We can also do this by using the number’s prime factors.

9. Writing 𝑎 in the form 𝑏 2 Example
Write 81 as a product of its prime factors and hence, show that 81 is a square number. 81
3 4 =81 ③ 27
2 =81
3×3 ③ 9 ③ ③
As we expressed 81 in the form 𝑎×𝑏×…×𝑥 2 , it must be square.

10. Writing 𝑎 in the form 𝑏 2 Example
Write 132 as a product of its prime factors and show that is not an integer.
132
2 2 ×3×11=132 ② 66 ② 33 ③ ⑪
We cannot write 2 2 ×3×11 in the form 𝑏 2 so 132 is not an integer.

11. Answers – Question 1 9 ② ② ③ ③ ② ② 3 2 =9 Square ③ ⑤
132
100 ② ② 66 50 ③ ③ ② 33 ② 25
3 2 =9 Square ③ ⑤ ⑪ ⑤
2 2 ×3×11≠ 𝑏 2 Not square
2 2 × 5 2 = Square
256
400 ②
128 ②
200
142 ② 64 ②
100 ② 71 ② 32 ② 50
2×71≠ 𝑏 2 Not square ②
2 8 = Square 16 ② 25 ② 8 ⑤ ⑤ ② 4
2 4 × 5 2 = Square ② ②

12. Answers – Question 2 999 289 225 ③ 75 ③ 333 ⑰ ⑰ ③ 25 ③ 111
17 2 =289 Integer ⑤ ③ 37 ⑤
3 2 × 5 2 = Integer
3 3 ×37≠ 𝑏 2 Not integer
6258
260 ②
3129 ②
130
784 ③
1043 ② 65 ②
392 ⑤ ②
196 ⑬ ⑦
149
2 2 ×5×13≠ 𝑏 2 Not integer ② 98 ② 49
2 4 × 7 2 = Integer ⑦ ⑦
2×3×7×149≠ 𝑏 2 Not integer