Presentation on theme: "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."— Presentation transcript:
1 The Square ExperimentSuppose 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 NUMBERSA 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 7Theorem: 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 prime2
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. 221e. 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 572: 2 x 2 x 2 x 3 x 312: 2 x x 3The 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 572: 2 x 2 x 2 x 3 x 3360: 2 x 2 x 2 x 3 x 3 x 5It 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) = 12lcm (60, 72) = 36012 x 360 = 432060 x 72 = 4320Is this a coincidence?
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 numbersChallenge problems Page 130 – #9 and 10Euclidean algorithm as time permits – following slides
19 If a = bq + r, then gcf(a,b) = gcf(b,r) Proof(outline): r = a – bqSo 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 tryFind gcf(348,72)Find gcf(78, 708)Find lcm(78,708)