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5.1 Divisibility

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Natural Numbers The set of natural numbers or counting numbers is {1,2,3,4,5,6,…}

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Factors The factors of a number are numbers that are multiplied together to equal that number. Example: What are the factors of 12? So the factors of 12 are 1, 2, 3, 4, 6, & 12. If 12 is divided by any of its factors the remainder is zero.

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Divisibility We say a is divisible by b if dividing a by b leaves a remainder of 0. We say that b is a divisor of a. Example: Since with no remainder we say that 24 is divisible by 8 8 divides 24 8 is a divisor of 24 We write 8|24

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Factors and divisibility Factors and divisors are the same. For example: 8 is a factor and divisor of 16 since

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Review the rules of divisibility p. 144

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Examples 4,681,396 is divisible by 2 since 6 is even 5,931,471 is divisible by 3 since 5 + 9 + 3 + 1 + 4 + 7 + 1 = 30 is divisible by 3 4,865,924 is divisible by 4 since 4 | 24 954 is divisible by 6 since 2 | 954 and 3 | 954 30,385 is divisible by 5 since it ends in 5 or 0 593,777,832 is divisible by 8 since the 8|832 543,186 is divisible by 9 since 5 + 4 + 3 + 1 + 8 + 6= 27 is divisible by 9 35,780 is divisible by 10 since it ends in 0 614,608,176 is divisible by 12 since 3 and 4 divide it

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Prime Numbers A prime number is a number greater than 1 with only 2 divisors or factors; 1 and itself. Example: 2, 3, 5, 7, 11, 13, 17, … Activity: Sieve of Eratosthenes

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Composite Numbers A composite number is a number > 1 with a factor other than 1 and itself. For example: 4, 6, 8, 9, 10, 12, 14, 15,…

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Prime Factorization The prime factorization of a number is expressing it as a product of its prime factors.

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Factor Trees We can show prime factorization using a factor tree: 340 34 10 21725 So 340 =

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Write a factor tree for the following numbers 700 180 510

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Greatest Common Factor The Greatest Common Factor or GCF is the greatest divisor of all the numbers. To find: 1.Write the prime factorization of each number 2.Select factors that are common to each 3.Take the smallest power of each of the factors selected 4.Multiply

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Examples Find the GCF of 225 and 825 Find the GCF of 72 and 120

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Relatively Prime If two numbers share no common factors other than one then they are called relatively prime. Example: 35 and 12 are relatively prime since they share no common factors other than 1

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Least Common Multiple The Least Common Multiple is the smallest number divisible by all of the numbers. One way to find the LCM is to list all multiples of each number and circle the smallest common one Example: To find the LCM of 15 and 20 Multiples of 15: 15, 30, 45, 60, 75,… Multiples of 20: 20, 40, 60, 80,… The LCM of 15 and 20 is 60.

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2 nd Way to Find LCM 1.Write prime factorization of each number 2.Select every factor 3.Take the highest power of each factor 4.Multiply

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Example Find the LCM of 18 and 30 Find the LCM of 144 and 300 Find the LCM of 60 and 108

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HW: p. 200/1-10,25-68

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