Numbers Lesson 1: Factors and Multiples of Whole numbers
Todays Objectives:
· Students will be able to demonstrate an understanding of factors of whole numbers by determining the: prime factors, greatest common factor (GCF), least common multiple (LCM), square root, cube root, Including:
· Determine the prime factors of a whole number
· Explain why the numbers 0 and 1 have no prime factors
· Determine, using a variety of methods, the GCF, or LCM of a set of whole numbers, and explain the process
Number Sets
Numbers are divided up into several different sets:
___________ ___________ (N)
¡ ____________
¡ Often called the “counting” numbers
___________ ___________ (W)
¡ ____________
¡ Includes all of the natural number set
___________ (I or Z)
¡ ___________________or can be written as ________________
¡ Includes all of the natural and whole number sets
_____________ _________ (Q)
¡ Any number that can be written as a ratio (fraction)
¡ Terminating decimal numbers, repeating decimal numbers
¡ Includes all of the natural, whole, and integer number sets
_____________ _________ (Q’)
¡ Any non-terminating, non-repeating decimal number
¡ Examples: π, ℮
__________ _____________(ℝ)
¡ All of the above number sets
Summary of Number Sets
Real Numbers
Factors of Whole Numbers
A _______ __________ is a number that is a member of the set W:{0, 1, 2, 3, 4 ,5,….}. Notice that ___ is a whole number, but it is not a member of the set of _________ _____________, N: {1, 2, 3, 4, 5,….}.
_______ of a number are numbers that _________ together to make that number. For example: 6 and 4 are factors of 24 (6 x 4 = 24).
24 has the following factors:
Prime Numbers and Factors
Any whole number greater than one that has only two distinct factors (one, and itself) is called a _________ ____________
Example: 13 is prime because it’s only factors are 1 and 13.
Numbers greater than 1 that are not prime are called ____________ numbers.
When the factors of a number are also prime, they are called _________ ___________.
Example: 12 has prime factors 2 x 2 x 3. We call this the _________ ____________________ of 12.
1 and 0
The number 1 is not a prime number because it is not ___________ by any whole numbers other than itself
The number 0 is not prime because it does not have 2 ____________factors
Example)
a) List the whole number factors of 40
Solution:
b) List the prime factors of 40.
Solution:
c) Write the prime factorization of 40.
Solution:
Example)(You do)
a) List the first 10 prime numbers
b) For the number 72, list the whole number factors, prime factors and write the prime factorization.
Factor trees
A factor tree is a ___________ used to write the prime factorization of a prime number
Greatest Common Factor (GCF)
The greatest common factor of two or more whole numbers is the __________ whole number that is a _______ of two or more numbers.
Example) The GCF of 12 and 18 is 6.
The largest shared factor between these two number is ___, so 6 is the ____. Here are some techniques to finding the GCF of different numbers.
Example) Determine the GCF of 24, 72, and 180
Solution 1: _____ all the factors of the three numbers; choose the largest factor _________by all three.
Solution 2: ________ the numbers into products of ________ of prime factors. The GCF is the product of the common powers with the smallest _________ associated with each power.
The bases 2 and 3 are _________ to all three prime factorizations. Since the power with base 2 with the smallest exponent is 22 and the power with a base of 3 with the smallest exponent is 31, the GCF is:____________
Solution 3: _________ the numbers by common prime factors until all the _________ do not have a common prime factor. In the method shown, the quotients are written under the __________ (divided numbers).
The numbers 2, 6, and 15 do ___ have a common prime factor. The product of the common prime factors is the GCF. This is the product of the divisors in the ______ __________: 2, 2, and 3. Thus, the GCF is 2x2x3 = 12.
Example)(You do)
Determine the GCF of 48, 80, and 120.
Least Common Multiple (LCM)
The least common multiple of two or more whole numbers is the __________ whole number that is a __________ of two or more whole numbers.
For example, the least common multiple of 12 and 18 is ____ because it is the smallest whole number that is a multiple of both 12 and 18.
There are a few techniques to finding the LCM, as shown in the next example.
Example) Determine the LCM of 8, 12, and 30.
Solution 1: _____ the multiples of 8, 12, and 30 until a common multiple is found
The smallest number that is a multiple of 8, 12, and 30 is _____, so the LCM is 120.
Solution 2: _______ each number into products of powers of ______ __________. The LCM is the product of the common powers that have the largest exponents associated with them, along with any non-common powers.
The LCM is the product of the common powers with the largest exponents (23 and 31), along with the non-common power (51). Thus, the LCM is:_______________.
Solution 3: use a similar ___________ technique as we used for the GCF. This time, we keep dividing until _____ of the numbers in a row have a common prime factor.
The LCM is the product of the left column and the bottom row.
Example)(You do)
Determine the LCM of 16, 18, and 20.
Ø Homework: Pg. 140-141 # 3-8ace, 9, 11, 13, 17, 20
Ø Read: Section 3.2: Perfect Squares, Perfect Cubes, and their Roots