Download Chapter 3 Factors and Products Factors are numbers multiplied to

Chapter 3 ­ Factors and Products
­ Factors are numbers multiplied to get a product. Ex. 4 and 5 are factors that multiply to give you a product of 20.
­ A multiple is the product you get from multiplying a given number by a natural number. Ex. Some multiples of 4 when multiplied by the natural numbers 1, 2, 3, 4 and 5 are: 4, 8, 12, 16, 20
(4 x 1) (4 x 2) (4 x 3) (4 x 4) (4 x 5)
­ A prime number only has two factors, 1 and itself. Ex. 17 is a prime number with the factors 1 and 17. Numbers that have more than two factors are called composite numbers. Ex. 12 is a composite number because it has 1, 2, 3, 4, 6 and 12 as factors. Even though 12 is a composite number, it can be prime factored. Of the six factors that 12 has, two of them are prime numbers (2 and 3). We can therefore prime factor 12 into: 2 x 2 x 3
­ Every composite number can be expressed as a product of prime factors. Ex. 36 is a composite number with the factors 1, 2, 3, 4, 6, 9, 12, 18, 36. The prime factors in this list are 2 and 3. This means 36 can be prime factored to be: 2 x 2 x 3 x 3
­ The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
­ Write the prime factors of 100. The first strategy is to use a factor tree. Ex. 100
4
Break 100 up into two of its factors. Break those factors down to their prime factors. The prime factorization of 100 is: 2 x 2 x 5 x 5 or 22 x 52
25
2 2 5 5
­ The second strategy to prime factor 100 is to use repeated division by prime factors. Begin by dividing 100 by the first prime factor 2. Divide by this prime factor until it no longer works then continue dividing by the next prime factor until the quotient is 1.
Ex. 100 ÷ 2 = 50
50 ÷ 2 = 25 (done using 2, 3 doesn't work either, try 5)
25 ÷ 5 = 5
5 ÷ 5 = 1 (We are done!)
The prime factors of 100 are 2 x 2 x 5 x 5 or 22 x 52.
- GCF - greatest common factor. The biggest factor that two
numbers share is called the GCF. Ex. 12 and 16. The first stragegy
to determine the GCF of 12 and 16 is to make a list of the factors:
Number
12
16
Factors
1, 2, 3, 4, 6, 12
1, 2, 4, 8, 16
The GCF of 12 and 16 is 4.
- A second strategy is to prime factor both 12 and 16.
12
16
2x2x3
2x2x2x2
Both numbers have at least two 2's, so
the GCF is 2 x 2 which is 4.
- LCM - least common multiple. The least common multiple of
two numbers is the least number that is divisible by each number.
Ex. 6 and 8. The first strategy to find the LCM of 6 and 8 is to make
a list of the multiples.
Number
6
8
Multiples
6, 12, 18, 24, ... The LCM of 6 and 8 is 24.
8, 16, 24, ...
- A second strategy to find the LCM of 6 and 8 is to use prime
factorization and choose the highest power of each prime factor in
any list.
6
8
2x3
= 21 x 31
2 x 2 x 2 = 23
The greatest power of 2 in any list is 23 and the greatest power of 3
in any list is 31. The least common multiple is the product of the
greatest power of each prime factor: 23 x 31 = 8 x 3 = 24 (LCM)
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