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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.
1 12 12
2 6 12
3 4 12
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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
24 8 3
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Factors and divisibility
• Factors and divisors are the same.
• For example:
8 is a factor and divisor of 16 since
2 8 16 and 16 8=2
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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
2 17 2 5
So 340 = 22 2 5 17 2 5 17
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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
number2. Select factors that are common to each3. Take the smallest power of each of the
factors selected4. 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 20Multiples 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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2nd 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