The student will

a.    find common multiples and factors, including least common multiple and greatest common factor.
b.    identify and describe prime and composite numbers; and identify and describe the characteristics of even and odd integers.

Number theory is a part of mathematics in which characteristics of numbers and relationships between numbers are examined.

The set of natural numbers is the set of counting numbers.
                  1, 2, 3, 4, 5, ...

The set of whole numbers is the set of natural numbers and zero.
                    0, 1, 2, 3, 4, ...

The set of integers is the set of whole numbers and their opposites.
                   ... -2, -1, 0, 1, 2, ...

An even integer is an integer that is divisible by 2.  

Ex. 2,4,6,8,10... are all even integers because they are divisible by 2   

An odd integer is an integer that is NOT divisible by 2.     

Ex. 1,3,5,7,9... are all odd integers because they are NOT divisible by 2  

A prime number is a natural number that has exactly two different factors, one and the number itself.

A composite number is a natural number that has more than two different factors.

One is neither a prime nor composite because it has only one factor, itself.

Zero is not a natural number. Although it has an infinite number of factors, zero is neither prime nor composite.

Prime Time Numbers "Song"   ~   Listen to Prime Time Numbers "Song"

Prime Number Checker

Teachers who have access to Discovery Streaming can show:

Investigating Prime Numbers 
Part Two: Prime Factorization from the Pumped-up World of Pre-Algebra
There are also some good clips from TEAMS

Prime or composite numbers can be represented by rectangular models or rectangular arrays on grid paper.

A prime number can be represented by only one rectangular array,
       7 can only be represented by a 7 × 1 array

A composite number can always represented by more than one rectangular array,
       9 can be represented by a 9 × 1 or a 3 × 3 array.

               

Prime or Composite Practice

How is the prime factorization of a composite number determined?

The prime factorization of a number is a representation of the number as the product of its prime factors.

The prime factorization of 18 is 2 × 3 × 3.  

Still not sure?  Check this out     ~     Prime Factorization Practice

Divisibility rules are useful tools in identifying prime and composite numbers.

Tests for Divisibility

Look at the digit in the one's place!

2 - any number that ends in an even digit (0, 2,4,6,8) is divisible by 2.

5 - if the last digit is 0 or 5, then the number is divisible by 6.

Check the sum of the digits!

3 - if the sum of the digits is divisible by 3, then the number is divisible by 3.

9 - if the sum of the digits is divisible by 9, then the number is divisible by 9.

Use your investigation skills!

6 - any even number that is divisible by 3, is also divisible by 6.

4 - if the last two digits in the number is divisible by 4, then the number is divisible by 4.

                       Divisibility Rules "Song"   ~   Listen to the Divisibility Words "Song"

Divisibility Rules Practice

A multiple of a number is the product of the number and any natural number.

The least common multiple of two or more numbers is the smallest common multiple of two or more numbers (other than 0).

The easiest way to find the LCM of two numbers is to use the List Method.  To start, 

1.    List the first five - ten nonzero multiples.

2.    Find the smallest number in common.  (It cannot be zero!)

Ex. Find the LCM of 9 and 21

Multiples of 9 =    9, 18, 27, 36, 45, 54, 63

Multiples of 21 =  21, 42, 63

LCM of 9 and 21 is 63 because it is the smallest multiple that is in both lists.

LCM Practice   ~    More Practice with Least Common Multiples

A factor of a number is an integer that divides evenly into that number. In other words it is a divisor of the number.

A common factor of two or more numbers is a divisor that all of the numbers share.

The greatest common factor of two or more numbers is the largest of the common factors.

Use Factor Pairs -

1.    List all of the factors of each number from least to greatest.

2.    The largest factor that is in both sets, is the greatest common factor.

Ex.  Find the GCF of 9 and 21

       Factors of 9 are: 1, 3, and 9

        Factors of 21 are: 1, 3, 7, 21

        GCF of 9 and 21 is 3 because 3 is the biggest number in both lists.

GCF Practice     ~     More Practice with Greatest Common Factor

Math 5 Live - Click on Skip Intro - Lessons - Multiples, Factors, Primes, and Composites

Spy Guys - Click on Skip Intro - Lessons - 7. Factors, Multiples, and Prime Factorization

More Practice - Scroll down to 6.3   ~   Sieve of Eratosthenes

TTAC Lessons from Enhanced Scope and Sequence

Partners and Leftovers     ~     Sieve of Eratosthenes     ~     LCMs and GCFs

Also check out the alternate lesson idea for Sieve of Eratosthenes!!

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