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

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.

Finding Primes Sieve of Erastothenes – Used to list primes. http://nlvm.usu.edu/en/nav/frames_asid_158_g_2_t_1.html?open=instructions

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

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

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. 327 d. 221 e. 337 f. 1000 g. 256

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.

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.

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

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.

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

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.

Find the prime factorization of: 60: 2 x 2 x 3 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.

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.

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

Theorem a·b = gcf(a,b)·lcm(a,b)

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

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

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

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