1. Factors and Multiples

2. What is a Prime Number?
A prime number has exactly two different divisors - one and itself.
Is one a prime number?
Zero?
Why or why not?
A composite number is a number with 3 or more factors.

3. Prime Numbers
“Prime numbers are what is left when you have taken all the patterns away. I think prime numbers are like life. They are very logical but you could never work out the rules, even if you spent all your time thinking about them.”
Mark Haddon, The Curious Incident of the Dog in the Night-time

4. Sieve of Eratosthenes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

5. Sieve of Eratosthenes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

6. Sieve of Eratosthenes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

7. Sieve of Eratosthenes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

8. Sieve of Eratosthenes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

9. Sieve of Eratosthenes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

10. I Claim...
The numbers left are prime numbers.
True or False, and why?

11. Prime Factors - Key Ideas
When a divisor of a number is a prime number, it is called a prime factor.
The prime factorization of a natural number is the number written as a product of its prime factors.
Every composite number can be expressed uniquely as a product of prime factors.

12. Fun Fact
A perfect number is one whose proper divisors sum to the number itself.
e.g. The number 6 has proper divisors 1, 2 and 3
= 6
28 has divisors 1, 2, 4, 7 and 14
= 28.

13. Factor tree
1980
198 10 22 9 5 2 11 2 3 3

14. Division by Prime Factors
1980 ÷ 2 = 990
990 ÷ 2 = 495
495 ÷ 3 = 165
165 ÷ 3 = 55
55 ÷ 5 = 11
11 ÷ 11 = 1
The prime factors of 1980 are 2, 3, 5 and 11
The prime factorization of 1980 is (2)(2)(3)(3)(5)(11) = (22)(32)(5)(11)

15. What letters do these words have in common?
sofa and mast as
scissors and stick
sic
kitten and knit
knit
Canada and banana
anaa

16. What if... We wrote Canada and banana as algebraic terms?
Canada = a3Cdn
banana = a3bn
The letters they have in common are aaan, or a3n
What about scissors and stick?
scissors = ciors4
stick = cikst
The letters they have in common are cis (notice they only have one factor of s in common, so we can only take one, not four)

17. Greatest Common Factor
The greatest common factor of two or more numbers is the greatest divisor the numbers have in common.
e.g. The factors of 8 are 1, 2, 4 and 8.
The factors of 12 are 1, 2, 3, 4, 6 and 12.
The greatest common factor of 8 and 12 is 4.
We write this as GCF(8, 12) = 4.

18. List Method List all the factors of each number.
Find all the factors common to all numbers.
The largest factor is the GCF.

19. List Method Find the GCF of 12 and 45. Find GCF(18, 54)
What are the advantages of this method? Disadvantages?

20. Try This...
Find a partner. How many push-ups can you do in one minute? Time each other.
Find the prime factorization of your own number.
What do you have in common? Use the lowest power of each prime.
Multiply the common primes together. Is this number a factor of both numbers?

21. Prime Factorization Method
Find the prime factorization of each number
Find the lowest power of each factor
Multiply the lowest power of each factor together

22. Prime Factorization Method
12 = (22)(3)
45 = (32)(5)
No common factor of 2
Lowest power of the common factor of 3 = 1
No common factor of 5
GCF (12, 45) = 3

23. Prime Factorization Method
GCF (28, 98)
GCF (32, 80)
GCF (27, 36,126)
What are the advantages and disadvantages to this method?

24. Example GCF Word Problem
A chocolate company has developed a new chocolate bar. They have decided that it should be 110 mm by 275 mm, and want to divide it evenly into the largest squares possible. How big can the squares be?

25. Une question pour vous
I’m knitting a sweater that has two different patterns. One pattern repeats every 12 rows, and the other every What is the minimum number of rows the sweater needs to be for both patterns to be finished at the end?

26. Lowest Common Multiple
The least common multiple of two or more numbers is the least number that is divisible by each number
e.g. Some multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108
Some multiples of 20 are 20, 40, 60, 80, 100, 120
The least common multiple is 60
We write this as LCM (12, 20) = 60
Lowest common denominator, Elimination

27. Multiple List Method List multiples each number
How far do you need to list?
Find the lowest multiple common to all
Find LCM (6, 22)
LCM (18, 20, 30)
Advantages? Disadvantages?

28. Prime Factorization Method
WAIT!!! We did a prime factorization method for GCF!!!
How did we find GCF with prime factorization?
LCM is the opposite
Instead of finding taking the smallest powers, we take the largest

29. Prime Factorization Method
Find LCM (6, 22)
24 = 23 3
54 = 2 33
LCM (6, 22) = = 216
Advantages? Disadvantages?

30. GCF vs. LCM How are they different? How are they the same? GCF LCM
Compare - How are they the same?
Contrast - How are they different?
How is each one used?
Examples