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§ 4-2 Prime and Composite Numbers Definitions What makes a whole number prime? Definitions What makes a whole number prime? Definition A prime number is a whole number that has exactly two distinct factors, one and itself. Definitions What makes a whole number prime? Definition A prime number is a whole number that has exactly two distinct factors, one and itself. If a number is not prime, then it is ... Definition A composite number is a whole number that has more than two distinct factors. Definitions What makes a whole number prime? Definition A prime number is a whole number that has exactly two distinct factors, one and itself. If a number is not prime, then it is ... Definition A composite number is a whole number that has more than two distinct factors. Our goal is to be able to either identify a whole number as prime or break the number up into the product of prime powers. We call this the prime factorization. Prime Factorization Example Find the prime factorization of 24. Prime Factorization Example Find the prime factorization of 24. 24 Prime Factorization Example Find the prime factorization of 24. 24 4 6 Prime Factorization Example Find the prime factorization of 24. 24 4 6 Prime Factorization Example Find the prime factorization of 24. 24 6 4 2 2 2 3 Prime Factorization Example Find the prime factorization of 24. 24 6 4 2 2 2 3 So, the prime factorization of is given by 24 = 23 · 3. Unique Factorization? Example Find the prime factorization of 24 by starting with different factors. Unique Factorization? Example Find the prime factorization of 24 by starting with different factors. 24 Unique Factorization? Example Find the prime factorization of 24 by starting with different factors. 24 3 8 Unique Factorization? Example Find the prime factorization of 24 by starting with different factors. 24 3 8 Unique Factorization? Example Find the prime factorization of 24 by starting with different factors. 24 3 8 Unique Factorization? Example Find the prime factorization of 24 by starting with different factors. 24 3 8 2 4 Unique Factorization? Example Find the prime factorization of 24 by starting with different factors. 24 3 8 2 4 Unique Factorization? Example Find the prime factorization of 24 by starting with different factors. 24 3 8 2 4 Unique Factorization? Example Find the prime factorization of 24 by starting with different factors. 24 3 8 2 4 2 2 Unique Factorization? Example Find the prime factorization of 24 by starting with different factors. 24 3 8 2 4 2 2 Fundamental Theorem of Arithmetic Is this unique factorization a fluke or is this expected? Fundamental Theorem of Arithmetic Is this unique factorization a fluke or is this expected? The Fundamental Theorem of Arithmetic The prime factorization of any whole number is unique, up to the order of the factors. Examples Example Find the prime factorization of each of the following whole numbers. 100 Examples Example Find the prime factorization of each of the following whole numbers. 100 = 22 · 52 Examples Example Find the prime factorization of each of the following whole numbers. 100 = 22 · 52 Because each factor is raised to the same power, we call this a perfect square. Examples Example Find the prime factorization of each of the following whole numbers. 100 = 22 · 52 Because each factor is raised to the same power, we call this a perfect square. We can also think of perfect squares as having an odd number of factors. Examples Example Find the prime factorization of each of the following whole numbers. 100 = 22 · 52 Because each factor is raised to the same power, we call this a perfect square. We can also think of perfect squares as having an odd number of factors. 3250 Examples Example Find the prime factorization of each of the following whole numbers. 100 = 22 · 52 Because each factor is raised to the same power, we call this a perfect square. We can also think of perfect squares as having an odd number of factors. 3250 = 2 · 53 · 13 Examples Example Find the prime factorization of each of the following whole numbers. 100 = 22 · 52 Because each factor is raised to the same power, we call this a perfect square. We can also think of perfect squares as having an odd number of factors. 3250 = 2 · 53 · 13 6468 Examples Example Find the prime factorization of each of the following whole numbers. 100 = 22 · 52 Because each factor is raised to the same power, we call this a perfect square. We can also think of perfect squares as having an odd number of factors. 3250 = 2 · 53 · 13 6468 = 22 · 3 · 72 · 11 Examples Example Find the prime factorization of 14053. Examples Example Find the prime factorization of 14053. Where do we start? Examples Example Find the prime factorization of 14053. Where do we start? Since none of our tests work, how far do we have to go with checking primes? Examples Example Find the prime factorization of 14053. Where do we start? Since none of our tests work, how far do we have to go with checking primes? √ 14053 ≈ 118.5 ⇒ 113 Examples Example Find the prime factorization of 14053. Where do we start? Since none of our tests work, how far do we have to go with checking primes? √ 14053 ≈ 118.5 ⇒ 113 14053 = 13 · 1081 Examples Example Find the prime factorization of 14053. Where do we start? Since none of our tests work, how far do we have to go with checking primes? √ 14053 ≈ 118.5 ⇒ 113 14053 = 13 · 1081 Now how far do we need to check? Examples Example Find the prime factorization of 14053. Where do we start? Since none of our tests work, how far do we have to go with checking primes? √ 14053 ≈ 118.5 ⇒ 113 14053 = 13 · 1081 Now how far do we need to check? √ 1081 ≈ 32.87 → 31 Examples Example Find the prime factorization of 14053. Where do we start? Since none of our tests work, how far do we have to go with checking primes? √ 14053 ≈ 118.5 ⇒ 113 14053 = 13 · 1081 Now how far do we need to check? √ 1081 ≈ 32.87 → 31 14053 = 13 · 23 · 47 Factorization Theorems Theorem If d is a divisor of n, then n d is a divisor of n. Factorization Theorems Theorem If d is a divisor of n, then n d is a divisor of n. Theorem If n is composite, then n has a prime factor p such that p2 ≤ n. Factorization Theorems Theorem If d is a divisor of n, then n d is a divisor of n. Theorem If n is composite, then n has a prime factor p such that p2 ≤ n. Theorem If n is a whole number greater than 1 and is not divisible by any prime such that p2 ≤ n, then n is prime. How Many Primes? How many prime numbers are there? How Many Primes? How many prime numbers are there? We think there are an infinite number, but ... How Many Primes? How many prime numbers are there? We think there are an infinite number, but ... If p1 , p2 , . . . , pk are a finite list of all primes, then what is p1 · p2 · . . . · pk + 1 How Many Factors? Example List all of the factors of 24. How Many Factors? Example List all of the factors of 24. The factors are {1, 2, 3, 4, 6, 8, 12, 24}. How Many Factors? Example List all of the factors of 24. The factors are {1, 2, 3, 4, 6, 8, 12, 24}. Do you notice anything about the factorizations of the factors in terms of 2 and 3? How Many Factors? Example List all of the factors of 24. The factors are {1, 2, 3, 4, 6, 8, 12, 24}. Do you notice anything about the factorizations of the factors in terms of 2 and 3? 1 = 20 · 30 How Many Factors? Example List all of the factors of 24. The factors are {1, 2, 3, 4, 6, 8, 12, 24}. Do you notice anything about the factorizations of the factors in terms of 2 and 3? 1 = 20 · 30 2 = 21 · 30 How Many Factors? Example List all of the factors of 24. The factors are {1, 2, 3, 4, 6, 8, 12, 24}. Do you notice anything about the factorizations of the factors in terms of 2 and 3? 1 = 20 · 30 2 = 21 · 30 3 = 20 · 31 How Many Factors? Example List all of the factors of 24. The factors are {1, 2, 3, 4, 6, 8, 12, 24}. Do you notice anything about the factorizations of the factors in terms of 2 and 3? 1 = 20 · 30 2 = 21 · 30 3 = 20 · 31 4 = 22 · 30 How Many Factors? Example List all of the factors of 24. The factors are {1, 2, 3, 4, 6, 8, 12, 24}. Do you notice anything about the factorizations of the factors in terms of 2 and 3? 1 = 20 · 30 2 = 21 · 30 3 = 20 · 31 4 = 22 · 30 6 = 21 · 31 How Many Factors? Example List all of the factors of 24. The factors are {1, 2, 3, 4, 6, 8, 12, 24}. Do you notice anything about the factorizations of the factors in terms of 2 and 3? 1 = 20 · 30 2 = 21 · 30 3 = 20 · 31 4 = 22 · 30 6 = 21 · 31 8 = 23 · 30 How Many Factors? Example List all of the factors of 24. The factors are {1, 2, 3, 4, 6, 8, 12, 24}. Do you notice anything about the factorizations of the factors in terms of 2 and 3? 1 = 20 · 30 2 = 21 · 30 3 = 20 · 31 4 = 22 · 30 6 = 21 · 31 8 = 23 · 30 12 = 22 · 31 How Many Factors? Example List all of the factors of 24. The factors are {1, 2, 3, 4, 6, 8, 12, 24}. Do you notice anything about the factorizations of the factors in terms of 2 and 3? 1 = 20 · 30 2 = 21 · 30 3 = 20 · 31 4 = 22 · 30 6 = 21 · 31 8 = 23 · 30 12 = 22 · 31 24 = 23 · 31 How Many Factors? Example How many factors does 36 have? How Many Factors? Example How many factors does 36 have? The prime factorization of 36 is How Many Factors? Example How many factors does 36 have? The prime factorization of 36 is 22 · 32 . How Many Factors? Example How many factors does 36 have? The prime factorization of 36 is 22 · 32 . 36 has how many factors? How Many Factors? Example How many factors does 36 have? The prime factorization of 36 is 22 · 32 . 36 has how many factors? 9 How Many Factors? Example How many factors does 36 have? The prime factorization of 36 is 22 · 32 . 36 has how many factors? 9 Pattern? How Many Factors? Example How many factors does 36 have? The prime factorization of 36 is 22 · 32 . 36 has how many factors? 9 Pattern? Pattern based on factorization? How Many Factors? Example How many factors does 36 have? The prime factorization of 36 is 22 · 32 . 36 has how many factors? 9 Pattern? Pattern based on factorization? Number of Factors If n = pn11 · pn22 · . . . · pnk k , then n has (n1 + 1)(n2 + 1) · · · (nk + 1) distinct factors. Listing Factors from Factorization Example List all of the factors of 45. Listing Factors from Factorization Example List all of the factors of 45. Product of factors Factor Listing Factors from Factorization Example List all of the factors of 45. Product of factors 30 · 50 Factor 1 Listing Factors from Factorization Example List all of the factors of 45. Product of factors 30 · 50 31 · 50 Factor 1 3 Listing Factors from Factorization Example List all of the factors of 45. Product of factors 30 · 50 31 · 50 32 · 50 Factor 1 3 9 Listing Factors from Factorization Example List all of the factors of 45. Product of factors 30 · 50 31 · 50 32 · 50 30 · 51 Factor 1 3 9 5 Listing Factors from Factorization Example List all of the factors of 45. Product of factors 30 · 50 31 · 50 32 · 50 30 · 51 31 · 51 Factor 1 3 9 5 15 Listing Factors from Factorization Example List all of the factors of 45. Product of factors 30 · 50 31 · 50 32 · 50 30 · 51 31 · 51 32 · 51 Factor 1 3 9 5 15 45 Classifications by Factors Definition A divisor of n is proper if it is not equal to n. Classifications by Factors Definition A divisor of n is proper if it is not equal to n. Definition A whole number is perfect if it equals the sum of its proper factors. Classifications by Factors Definition A divisor of n is proper if it is not equal to n. Definition A whole number is perfect if it equals the sum of its proper factors. Example 6 is a perfect number because 1 + 2 + 3 = 6. Classifications by Factors Definition A divisor of n is proper if it is not equal to n. Definition A whole number is perfect if it equals the sum of its proper factors. Example 6 is a perfect number because 1 + 2 + 3 = 6. What is the next perfect number? Classifications by Factors Definition A divisor of n is proper if it is not equal to n. Definition A whole number is perfect if it equals the sum of its proper factors. Example 6 is a perfect number because 1 + 2 + 3 = 6. What is the next perfect number? 28 Classification by Factors While searching for 28, did you find any whole numbers whose sum of proper factors exceeds the whole number itself? Classification by Factors While searching for 28, did you find any whole numbers whose sum of proper factors exceeds the whole number itself? Definition A whole number is abundant if it is larger than the sum of its proper factors. Classification by Factors While searching for 28, did you find any whole numbers whose sum of proper factors exceeds the whole number itself? Definition A whole number is abundant if it is larger than the sum of its proper factors. Any you found where the sum of proper factors was less than the whole number? Classification by Factors While searching for 28, did you find any whole numbers whose sum of proper factors exceeds the whole number itself? Definition A whole number is abundant if it is larger than the sum of its proper factors. Any you found where the sum of proper factors was less than the whole number? Definition A whole number is deficient if it is smaller than the sum of its proper factors.