greatest common factor. what is a greatest common factor? the greatest common factor (gcf) of a...
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Greatest Common FactorGreatest Common Factor
Greatest Common FactorGreatest Common Factor
What is aWhat is aGreatest Common Factor?Greatest Common Factor?
The greatest common factor (GCF) of a group of numbers is the largest (greatest)
of all the factors the numbers have in common.
Example:
What is greatest common factor of the following group?
8, 12, 20
Though they have factors of 1, 2 and 4 in common,4 is the GCF4 is the GCF
Greatest Common FactorGreatest Common Factor
When would we need to knowWhen would we need to knowwhat the GCF of a group of numbers is?what the GCF of a group of numbers is?
There are many times we want to know the GCF of a group of numbers.
Here are a few:
Reducing FractionsReducing Fractions
You can completely reduce a fraction in 1 stepif you divide the numerator and denominator
by the greatest of their common factors – the GCF!
Greatest Common FactorGreatest Common Factor
When would we need to knowWhen would we need to knowwhat the GCF of a group of numbers is?what the GCF of a group of numbers is?
There are many times we want to know the GCF of a group of numbers.
Here are a few:
Equal DivisionsEqual Divisions
Say you have 3 pieces of wood…their sizes are:24 ft. 12 ft. 9 ft.
You want to cut the wood into smaller pieces of the same size, but you want the pieces to be a big as possible, without any waste. How big should you cut them?
Greatest Common FactorGreatest Common Factor
How do you find a GCF?How do you find a GCF?
There are a number of ways to find a GCF of a group of numbers.
Method 1:Method 1:
List all the factors of each number, and pick out the greatest of all the factors they have in common.
Example:
Find the GCF of 9, 12, 24Find the GCF of 9, 12, 24
9: 1, 3, 9 12: 1, 2, 3, 4, 6, 12 24: 1, 2, 3, 4, 6, 8, 12, 249: 1, 3, 9 12: 1, 2, 3, 4, 6, 12 24: 1, 2, 3, 4, 6, 8, 12, 24
They all have factors of 1 and 3. They all have factors of 1 and 3.
3 would be the GCF.3 would be the GCF.
Greatest Common FactorGreatest Common Factor
How do you find a GCF?How do you find a GCF?
There are a number of ways to find a GCF of a group of numbers.
Method 2:Method 2:
The second method of finding the GCF of a group of numbers is to use their Prime Factorization.
The GCF of a group of numbers isThe GCF of a group of numbers isthe product of ALL their common prime factors.the product of ALL their common prime factors.
Greatest Common FactorGreatest Common Factor
Finding the GCF using the Prime Factorization.Finding the GCF using the Prime Factorization.
The GCF of a group of numbers isThe GCF of a group of numbers isthe product of ALL their common prime factors.the product of ALL their common prime factors.
Example:
Find the GCF of: 24, 36, 60
1. We need the prime factorizations
32 3 2 22 3 22 3 5
Greatest Common FactorGreatest Common Factor
Finding the GCF using the Prime Factorization.Finding the GCF using the Prime Factorization.
The GCF of a group of numbers isThe GCF of a group of numbers isthe product of ALL their common prime factors.the product of ALL their common prime factors.
Example:
Find the GCF of: 24, 36, 60
2. Now we need to multiply all the common prime factors
What do they all have in common?
They all have two 2s in common.They all also have one 3 in common.
324 2 3 2 236 2 3 260 2 3 5
2GCF 2 3 4 3 12
Greatest Common FactorGreatest Common Factor
Which method always works?Which method always works?
BOTH!
Which method is better?Which method is better?
Depends on the numbers…
With small numbers, listing the factors is probably easier
With larger numbers,using the prime factorization is probably easiest
Greatest Common FactorGreatest Common Factor
Let's Practice…Let's Practice…
Use the prime factorization to find the GCF of the following group of numbers.
48, 72
448 2 3 3 272 2 3
Greatest Common FactorGreatest Common Factor
Let's Practice…Let's Practice…
Use the prime factorization to find the GCF of the following group of numbers.
48, 72448 2 3 3 272 2 3
3GCF 2 3 8 3 24
Greatest Common FactorGreatest Common Factor
Let's Practice…Let's Practice…
Use the prime factorization to find the GCF of the following group of numbers.
27, 64
327 3 664 2
Greatest Common FactorGreatest Common Factor
Let's Practice…Let's Practice…
Use the prime factorization to find the GCF of the following group of numbers.
27, 64
GCF ? 1
327 3 664 2
Greatest Common FactorGreatest Common Factor
Special Groups of NumbersSpecial Groups of Numbers
If the GCF of a group of numbersis 1, then we call the group…
RELATIVELY PRIMERELATIVELY PRIME
A group of numbers is "relatively prime"if they have no common factors other than 1.
Can you think of a group of numbers thatCan you think of a group of numbers thatwould be considered relatively prime?would be considered relatively prime?
Greatest Common FactorGreatest Common Factor
Which group(s) of numbers are relatively prime?Which group(s) of numbers are relatively prime?
4, 12, 20
3, 9, 18, 23
9, 72
2, 3, 4
Greatest Common FactorGreatest Common Factor
More Practice…More Practice…
Use the prime factorization to find the GCF of the following group of numbers.
128 & 96
360 & 540
7128 2596 2 3 5GCF 2 32
3 2360 2 3 5 2 3540 2 3 5
2 2GCF 2 3 5 180