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

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

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

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

Answers – Questions 7 - 12 150 770 85 ② 75 ② 385 ⑤ ⑰ ③ 25 ⑤ 77 5×17=85 ⑦ 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

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 16 produces an integer. As 16 = 2, 16 must be an integer ( 16 =± ). 4 4 We can also do this by using the number’s prime factors.

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.

Writing 𝑎 in the form 𝑏 2 Example Write 132 as a product of its prime factors and show that 132 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.

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 = 10 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 = 16 2 Square 16 ② 25 ② 8 ⑤ ⑤ ② 4 2 4 × 5 2 = 20 2 Square ② ②

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 = 15 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 = 28 2 Integer ⑦ ⑦ 2×3×7×149≠ 𝑏 2 Not integer