7.5 Prob. Solving: Make an Organized List (p.174-175)

When solving story problems lists can help you:

• recognize patterns
• find common multiples or factors
• solve multistep problems

7.4 Greatest Common Factor- GCF (P.170-173)

I. There are two methods for finding the greatest common factor:

• List all factors and find the largest
• multiply the common prime factors

A.List and find largest
1.First, list all of the factors of each number
2.Then, list the common factors and choose the largest one.

B.List the prime factors, then multiply the common prime factors.
1. Find the prime factors for the numbers in question
2.Recognize the common prime factors
3. Multiply them together to find the greatest common factor

Example for list all factors:
Find the GCF of 36 and 54.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18

Although the numbers in bold are all common factors of both 36 and 54, 18 is the greatest common factor.

Example for multiplying common prime factors:
Let's use the same numbers, 36 and 54 again to find their greatest common multiple.
The prime factorization of 36 is 2 x 2 x 3 x 3
The prime factorization of 54 is 2 x 3 x 3 x 3


Notice that the prime factorizations of 36 and 54 both have one 2 and two 3s in common. So, we simply multiply these common prime factors to find the greatest common factor. Like this...

2 x 3 x 3 = 18

Both methods for finding the greatest common factor work!

-examples from helpwithfractions.com

7.4 Least Common Multiple (p.170-173) USING PRIME FACTORS

When working to find the LEAST COMMON MULTIPLE (LCM) with large or inconvenient numbers use the prime factors.

1.Find the prime factors for all numbers involved.

2.Count the number of times each prime number appears in each of the factorizations.

3.For each prime number, take the largest of these counts.

4.Write down that prime number as many times as you counted for it in step 3.

5.The least common multiple is the product of all the prime numbers written down.

Example: Find the least common multiple of 5, 6 and 15.
•Factor into primes
Prime factorization of 5 is 5
Prime factorization of 6 is 2 x 3
Prime factorization of 15 is 3 x 5
•Notice that the different primes are 2, 3 and 5.
•Now, we do Step #2 - Count the number of times each prime number appears in each of the factorizations...
The count of primes in 5 is one 5
The count of primes in 6 is one 2 and one 3
The count of primes in 15 is one 3 and one 5
•Step #3 - For each prime number, take the largest of these counts. So we have...
The largest count of 2s is one
The largest count of 3s is one
The largest count of 5s is one
•Step #4 - Since we now know the count of each prime number, you simply - write down that prime number as many times as you counted for it in step 2.
Here they are...
2, 3, 5
•Step #5 - The least common multiple is the product of all the prime numbers written down.
2 x 3 x 5 = 30

•Therefore, the least common multiple of 5, 6 and 15 is 30.

7.4 Least Common Multiple (p.170-173)

Least Common Multiple (LCM)
I. A multiple is the product of a whole number when multiplied by another whole number.
A. For example: 9x1=9,9x2=18,9x3=27,9x4=36, etc... so 9, 18, 27, 36, etc... are multiples of 9.

II. How to find the Least Common Multiples (LCM)
A. Make a list of multiples for the numbers in question
B. Continue the list until their is a common multiple between the two
C. Identify the lowest shared multiple between the shared numbers

III. Tips to finding the LCM
A. Start by making a list for the larger of the two values in question and use your math facts to check in you head if the smaller number is a factor of any of the multiples you are listing
B. Don't forget that the original number is a multiple (multiplied by 1)
C. If you continue your list you will find more factors, but not the "least"

Below is a video example from SchoolTube.com

7.3 Prime Factorization (p.168-169)

I. Use a factor tree to find the prime factors
A. Steps for prime factorization

1. Choose any two factors (other than the 1) for the number in question. Draw two lines, like branches, down from the original number and write the factors at the ends of the lines.
2. Check the numbers at the ends of the branches to see if they are prime or comoposite numbers. If they are prime, circle them to include as prime factors.
3. Draw two more branches out of all composite factors until you only have prime factors at the ends of the branches.

B. You will always end up with the same prime factors (not necessarily in the same order), regardless of the composite factors you start with.

-Below are two different examples of how you could find the prime factors for the number 700.

7.2 Prime and Composite Numbers

I Prime Number
- a whole number that only has two factors: 1 and itself

II. Composite Number
- a whole number that has at least three factors
- All even numbers, except for 2, are composite numbers

***0 and 1 are neither prime or composite

Scroll down on this site from AAAmath.com to take a quiz on prime and composite numbers

7.1 Divisibility

Divisibility Rules:

1. All numbers are divisible by 1
2. If the last digit is even (0,2,4,6,or8) it is divisble by 2
3. If the sum of the digits is divisible by 3, then so is the original number
4. If the last two digits form a number divisible by 4
5. If the last digit is 0 or 5
6. If the number is divisible by 2 and 3
7. Double the last digit and subtract it from the rest of the digits. (Repeat the process
   for larger numbers.) If you get a # that is divisible by 7, then so is the number.
8. If the last three digits form a number divisible by 8
9. If the sum of the digits is divisible by 9
10. If the last digit is 0

Test what you know by applying the divisibility rules for 10 practice numbers at vectorkids.com

Chapter 7 Video Tutorial

Here is a rather lengthy video that covers most of the materials in chapter 7.