Download 5.1 Divisibility

5.1 Divisibility
Natural Numbers
• The set of natural numbers or counting
numbers is {1,2,3,4,5,6,…}
Factors
• The factors of a number are numbers that
are multiplied together to equal that number.
• Example: What are the factors of 12?
112  12
2  6  12
3 12
4 are
12 1, 2, 3, 4, 6, & 12. If
So the factors of
12 is divided by any of its factors the
remainder is zero.
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 24  8  3 with no remainder we say that
24 is divisible by 8
8 divides 24
8 is a divisor of 24
We write 8|24
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
Review the rules of divisibility
p. 144
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
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
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,…
Prime Factorization
• The prime factorization of a number is
expressing it as a product of its prime
factors.
Factor Trees
• We can show prime factorization using a
factor tree:
340
34
2
10
17
2
So 340 = 2  2  5 17  2  5 17
2
5
Write a factor tree for the
following numbers
• 700
• 180
• 510
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
Examples
• Find the GCF of 225 and 825
• Find the GCF of 72 and 120
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
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.
nd
2
1.
2.
3.
4.
Way to Find LCM
Write prime factorization of each number
Select every factor
Take the highest power of each factor
Multiply
Example
• Find the LCM of 18 and 30
• Find the LCM of 144 and 300
• Find the LCM of 60 and 108
HW: p. 200/1-10,25-68