Transcript The factors of 8

Finding Roots
Composite Numbers
STEP 1:
Find all the factors of a number
What are factors and how do I find them?
Example: Say you want to find the factors of 8.
The factors of 8 are all the numbers that will divide into 8 evenly.
(In other words, they are not decimals like 2.38 or 4.1)
So take the numbers from 1 to 8 and divide 8 by each of them.
8
1) 8
4
2) 8
decimal
3) 8
2
4) 8
If you get an answer with a decimal in it, the number you
divided by is not a factor of 8, so cross out these answers.
decimal
decimal
decimal
5) 8
6) 8
7) 8
Now let’s look at the numbers that are left
1
8) 8
The numbers on top are the factors of 8, factors: 8, 4, 2 and 1
8
1) 8
4
2) 8
2
4) 8
1
8) 8
But, did you notice that the numbers on the top and the
numbers you divided by (on the left) are the same?
That’s because we are finding the factors two at a time,
The number on the left and the number on top
are both factors of 8.
So to save time we don’t have to divide by every
number from 1 to 8, we can go halfway and stop.
If we only have to find half of the factors,
how do we know when we have gotten
halfway and can stop?
If you write the factors of the number using the following system,
you can see where your stopping point will be.
1) Write the number with two little branches below it
2) Starting with ‘1 x 8’
Write all the pairs of factors
that divide evenly into 8
3) This is where they
start to repeat, STOP HERE!
8
1*
2*
4*
8*
8
4
2
1
All the factors
of 8 are right
here in this
little box.
You don’t need to write these
repeating numbers down
Practice:
Find the factors of the following numbers
Read the
factors
in this
order
Down
the
left
side
12
1 * 12
2* 6
3* 4
We can stop
checking
numbers
as soon as
we reach
this number
Up
the
right
side
32
81
1 * 32
2 * 16
4* 8
1 * 81
3 * 27
9* 9
5 * decimal
6 * decimal
7 * decimal
8 * 4(repeat)
You can stop here
Here’s where the
since there are no
numbers start to
more numbers
between these two
repeat 4 * 3, etc.
factors on the bottom
so stop here.
Make sure you check all the numbers up to
the number on the bottom right, this is
where they start to repeat.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 32: 1, 2, 4, 8, 16, 32
48
1 * 48
2 * 24
3 * 16
4 * 12
6* 8
7 * decimal
Stop, since the
next number is 8
Factors of 81: 1, 3, 9, 27, 81
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
This is a square
root, so look for
perfect squares
Step 2: Split the number into two factors
Use the splitting property to simplify the following:
1) Find all the pairs of factors-look for perfect squares
2) Find the pair with the largest perfect square
1 * 12
2* 6
3* 4
3) Write this pair in the following order:
4 is a perfect square
Answer
4) Take the square root of the perfect number
This is a square
root, so look for
perfect squares
Step 2: Split the number into two factors
Use the splitting property to simplify the following:
1) Find all the pairs of factors-look for perfect squares
2) Find the pair with the largest perfect square
1 * 32
2 * 16
4* 8
3) Write this pair in the following order:
4 and 16 are perfect squares
Answer
4) Take the square root of the perfect number
This is a square
root, so look for
perfect squares
Step 2: Split the number into two factors
Use the splitting property to simplify the following:
1) Find all the pairs of factors-look for perfect squares
1 * 81
3 * 27
9* 9
2) Double factors like this mean that the original number
was a perfect square and this splitting process is
unnecessary.
3) Take the square root of 81 (see perfect numbers chart)
Answer
Note: Checking for Prime numbers should also be done before trying
the splitting process because prime numbers cannot be broken up at all.
Answer
This is a square
root, so look for
perfect squares
Step 2: Split the number into two factors
Use the splitting property to simplify the following:
1) Find all the pairs of factors-look for perfect squares
2) Find the pair with the largest perfect square
1*
2*
3*
4*
6*
48
24
16
12
8
3) Write this pair in the following order:
Answer
4 and 16 are perfect squares
4) Take the square root of the perfect number
This is a cube root,
so look for perfect
cubes
Step 2: Split the number into two factors
Use the splitting property to simplify the following:
1) Find all the pairs of factors-look for perfect cubes
2) Find the pair with the largest perfect cube
1 * 108
2 * 54
3 * 36
4 * 27
6 * 18
9 * 12
3) Write this pair in the following order:
27 is a perfect cube
Answer
4) Take the cube root of the perfect number
Practice Problems