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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

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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.

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Finding Primes Sieve of Erastothenes – Used to list primes.

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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

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

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How do we find prime factorizations? In groups – which of the following are prime. For those that are not, find the prime factorization. a.127b. 129c. 327 d. 221 e. 337f. 1000g. 256

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

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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.

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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.

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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.

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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

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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.

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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.

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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 dont include anything more.

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

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Theorem a·b = gcf(a,b)·lcm(a,b)

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Groupwork Find the gcf and lcm of each pair 1.30, , , , 9 5.p, q where p and q are different prime numbers 6.Challenge problems Page 130 – #9 and 10 Euclidean algorithm as time permits – following slides

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

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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 its also a factor of r. Consider: 1.The common factors of a and b. 2.The common factors of b and r. They are the same. So the greatest in each group is the same

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Examples to try 1.Find gcf(348,72) 2.Find gcf(78, 708) 3.Find lcm(78,708)

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