1. The Square Experiment
Suppose you have six squares tiles. How many different rectangles can you make?
How many could you make with 1 tile? 2 tiles? 3 tiles?
n tiles where n is any number from 1to 30

2. PRIME NUMBERS
A prime number is a whole number, P>1, whose only factors are 1 and P.
A whole number, N>1, which is not prime is called composite.
Note that 1 is neither prime nor composite.

3. Finding Primes Sieve of Erastothenes – Used to list primes.

4. Finding Primes Sieve of Erastothenes – Used to list primes.
Note – We only need to examine prime factors up to 7
Theorem: Let N be a whole number. Let k be the largest prime so that k < N. If none of the primes less than or equal to k are factors of N, then N is also prime 2

5. Fundamental Theorem of Arithmetic
Every whole number N > 1 can be written uniquely as a product of primes.

6. How do we find prime factorizations?
In groups – which of the following are prime. For those that are not, find the prime factorization.
127 b. 129 c d. 221
e. 337 f g. 256

7. In Groups What is mean by: the least common multiple of two numbers.
The greatest common divisor of two numbers.
Use 60 and 72 as examples of your two numbers.

8. Definition: The greatest common factor of two numbers, a and b, written gcf(a,b), is the greatest whole number which is a factor of both a and b. Example – Find gcf(60,72) by listing the factors of each.

9. Find the prime factorization of:
60: 2 x x x 5
72: 2 x 2 x 2 x 3 x 3
12: 2 x x 3
The gcf seems to consist of the prime factors that the two numbers have in common.

10. WHY?
If n is a factor of a, then all prime factors of n are also prime factors of a.
If n is a factor of b, then all prime factors of n are also prime factors of b.

11. So if n is a factor of both a and b, then all prime factors of n will be prime factors of both a and b.
To get the gcf(a,b), we need to find the largest such factor. It will be the one that includes all the factors that a and b have in common

12. Definition: The least common multiple of two numbers, a and b, written lcm(a,b), is the least whole number which is a multiple of both a and b. Example – Find lcm(60,72) by listing the multiples of each.

13. Find the prime factorization of:
60: 2 x x x 5
72: 2 x 2 x 2 x 3 x 3
360: 2 x 2 x 2 x 3 x 3 x 5
It seems that the lcm must include all the prime factors of each number.

14. WHY?
If n is a multiple of a, then all the prime factors of a are also prime factors of n.
If n is a multiple of b, then all the prime factors of a are also prime factors of b.
So to find the lcm(a,b), be sure include all the prime factors of a and b, but don’t include anything more.

15. ??????
gcf (60, 72) = 12
lcm (60, 72) = 360
12 x 360 = 4320
60 x 72 = 4320
Is this a coincidence?

16. Theorem
a·b = gcf(a,b)·lcm(a,b)

17. Groupwork Find the gcf and lcm of each pair 30, 42 72, 96 12, 132 4, 9
p, q where p and q are different prime numbers
Challenge problems Page 130 – #9 and 10
Euclidean algorithm as time permits – following slides

18. Euclidean Algorithm
If a = bq + r,
then gcf(a,b) = gcf(b,r)

19. If a = bq + r, then gcf(a,b) = gcf(b,r)
Proof(outline): r = a – bq
So if n is a factor of a and b, then it’s also a factor of r.
Consider:
The common factors of a and b.
The common factors of b and r.
They are the same. So the greatest in each group is the same

20. Examples to try
Find gcf(348,72)
Find gcf(78, 708)
Find lcm(78,708)