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# 1150 day 7

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### 1150 day 7

1. 1. Prime and CompositeNumbersGreatest Common Divisor &Least Common Multiple
2. 2. A Prime Number has exactly two distinct positivedivisors. A Prime Number can only be divided by itself and 1.Prime Numbers: 2 3 A composite number has factors 5 other than itself and 1. 7 11 Example: 6 is composite because 13 its factors are 1, 2, 3 and 6. 17 19 23 29…
3. 3. A factor tree can be used to find the prime factorsof a composite number. 24 24 8 3 6 4 2 4 2 3 2 2 2 2Prime factorization Fundamental Theorem of of 24: Arithmetic:24 = 2 · 2 · 2 · 3 Each composite number can be or written as a product of primes in24 = 23 · 3 one way only.
4. 4. Is 43 prime or composite? If 43 is composite, it has prime factorsDivide 43 by prime numbers to see if they are divisors 43 2 = 21 R 1 43 3 = 14 R 1 43 5 = 8 R 3 43 7 = 6 R 1 43 11 = 3 R 10 How far did we really have to go 43 is prime. before we could determine that 43 is prime?
5. 5. What is the largest possible prime factor of 43? 5 Prime Square 2 4 43 2 = 21 R 1 3 9 43 3 = 14 R 1 5 25 43 5= 8R3 43 7 49 43 7= 6R1 11 121 43 11 = 3 R 10 13 169 17 289
6. 6. Is 113 prime or composite? Prime Square 2 4 Is 2 a divisor of 113? No 3 9 Is 3 a divisor of 113? No 5 25 Is 5 a divisor of 113? No 7 49 Is 7 a divisor of 113? No 113 11 121 13 169 17 289 113 is prime
7. 7. Divisors of Composite NumbersList all divisors of 12 1, 2, 3, 4, 6, 12How many divisors does 1500 have? 1500 1500 = 22 · 31 · 53 15 100 Number of divisors: 3 5 10 10 (2 + 1)(1 + 1)(3 + 1) = (3)(2)(4) 2 5 2 5 = 24
8. 8. The Greatest Common Divisor (GCD) of two or moreintegers is the largest integer that divides the numbers.Find the GCD of 8 and 20Intersection of sets methodFactors of 8: 1, 2, 4, 8Factors of 20: 1, 2, 4, 5, 10, 20Common Factors: 1, 2, 4GCD(8, 20) = 4
9. 9. Find the GCD of 8 and 20Prime factorization method 8 20 8=2·2·2 4 2 4 520 = 2 · 2 · 5 2 2 2 2 2 2GCD(8, 20) = 2 · 2 = 4
10. 10. Find the GCD of 42 and 63Intersection of sets methodFactors of 42: 1, 2, 3, 6, 7, 14, 21, 42,Factors of 63: 1, 3, 7, 9, 21, 63Common Factors: 1, 3, 7, 21GCD(42, 63) = 21
11. 11. Find the GCD of 42 and 63Prime factorization method 42 6342 = 2 · 3 · 7 6 7 9 763 = 3 · 3 · 7 2 3 3 3 3 7GCD(42, 63) = 3 · 7 = 21
12. 12. Euclidean AlgorithmThe GCD(a, b) = GCD(r, b) where r is the remainderwhen a is divided by b.GCD(42, 63) 1 2 42 63 21 42 42 42 21 0 0 remainder GCD = 21
13. 13. GCD(1824, 7448) 4 12 1824 7448 152 1824 7296 152 152 304 304 0 0 remainder GCD = 152 Check: Is 152 a the GCD of 1824 and 7448? 1824 ÷ 152 = 12 7448 ÷ 152 = 49
14. 14. The Least Common Multiple (LCM) of two or morenumbers is the smallest number that is a multiple ofall the numbers.Find LCM(8, 20)Intersection of sets method:Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …Multiples of 20: 20, 40, 60, 80, …Common Multiples: 40, 80, 120, …LCM: 40
15. 15. Find LCM(8, 20) 8 20Prime factorization method 4 4 5 2 8=2·2·2 2 2 2 220 = 2 · 2 · 5 L C M 2 2 2 5 E O U F U L T N TLCM(8, 20) = 2 · 2 · 2 · 5 = 40 O T I V with P E L R E S S