##Factors
Factors are whole numbers that can evenly divide another number.
Picturing factors
Factors give us a way to break down a number into smaller pieces. We can arrange dots into equal sized groups to help us picture the factors of $12$.
$12$ dots can be arranged in $\tealC{1}$ row with $\tealC{12}$ dots.
$\tealC{1 \times 12} = 12$
[[☃ image 1]]
$12$ dots can also be arranged in $\purpleC{2}$ rows with $\purpleC{6}$ dots per row.
$\purpleC{2 \times 6} = 12$
[[☃ image 2]]
Or, we can arrange $12$ dots in $\redC{3}$ rows with $\redC{4}$ dots in each row.
$\redC{3 \times 4} = 12$
[[☃ image 3]]
[[☃ explanation 3]]
Once we figure out all of the ways that $12$ dots can be arranged, we can look at the number of rows and number of dots in each row to find the factors of $12$.
$\tealC{1}$, $\tealC{12}$, $\purpleC{2}$, $\purpleC{6}$, $\redC{3}$, and $\redC{4}$ are all factors of $12$.
We can make $12$ with a row of $\greenD5$ and a row of $\goldD7$. So are $\greenD5$ and $\goldD7$ factors of $12$?
[[☃ image 4]]
[[☃ image 5]]
No. $\greenD5$ and $\goldD7$ are not factors because the dots are not divided into equal sized groups.
[[☃ graded-group 3]]
[[☃ explanation 2]]
[[☃ graded-group 4]]
[[☃ explanation 4]]
###Finding factors without pictures
We can find the factors for $16$ without drawing dots by thinking about numbers that will divide into $16$ evenly.
$\blueD{1}$ is a factor of $16$ because $\blueD{1}$ can be divided into $16$ with no remainder.
$16 \div \blueD{1}= \blueD{16}$
The quotient, which is $\blueD{16},$ is also a factor of $16$.
$\greenD{2}$ is a factor of $16$ because $\greenD{2}$ can be divided into $16$ with no remainder.
$16 \div \greenD{2}= \greenD{8}$
The quotient, which is $\greenD{8}$, is also a factor of $16$.
$\purpleD{4}$ is a factor of $16$ because $\purpleD{4}$ can be divided into $16$ with no remainder.
$16 \div \purpleD{4}= \purpleD{4}$
In this case the quotient is $\purpleD{4}$, which we have already discovered is a factor of $16$.
The factors of $16$ are $\blueD{1, 16}$, $\greenD{2, 8},$ and $\purpleD{4}$.
Numbers like $3$ and $5$ are not factors of $16$ because they cannot be divided evenly into $16$.
[[☃ graded-group 2]]
[[☃ explanation 1]]
##Factor hints
Every number has $1$ as a factor.
$1$ is a factor of $10$.
$1$ is a factor of $364$.
$1$ is a factor of $5,787$.
Every number has itself as a factor.
$41$ is a factor of $41$.
$128$ is a factor of $128$.
$4,379$ is a factor of $4,379$.
##Factor pairs
Two numbers that we multiply together to get a certain product are called factor pairs. To get the product of $8$, we can multiply $\purpleE{1}$ $\times$ $\purpleE{ 8}$ and $\tealE{2}$ $\times$ $\tealE{4}$. So the factor pairs for $8$ are $\purpleE{1}$ and $\purpleE{8}$ and $\tealE{2}$ and $\tealE{4}$.
Arranging dots in equal sized groups helps us to see that factors always come in pairs. One factor in the factor pair is the number of rows. The other factor in the factor pair is the number of dots in each row.
[[☃ image 1]]
[[☃ image 2]]
Let's find the factor pairs of $20$. Remember, we are looking for two whole numbers that we can multiply together to get $20$.
We'll start with $\blueE{1}$ because we know that $\blueE{1}$ is a factor for every number.
We multiply $\blueE{1} \times \blueE{20}$, to get $20$, so $\blueE{20}$ is also a factor. We can list these factors as the outside ends of a list, leaving room in the middle for additional factors.
| | | | | |
- | - | - | - | - |
$\blueE{1}$| | | | | $\blueE{20}$
Now we check to see whether the next counting number, $2$, is a factor.
Is there a whole number we can multiply by $\tealE{2}$ to get $20$? Yes. $\tealE{2} \times \tealE{10}=20$. So $\tealE{2}$ and $\tealE{10}$ are another factor pair.
| | | | | |
- | - | - | - | - |
$\blueE{1}$| $\tealE{2}$ | | | $\tealE{10}$ | $\blueE{20}$
The next counting number is $3$. Is there a whole number we can multiply by $3$ to get $20$? No. So $3$ is not a factor of $20$.
Can we multiply $\redE{4}$ by a whole number to get $20$? Yes. $\redE{4} \times \redE{5}= 20$. So $\redE{4}$ and $\redE{5}$ are a factor pair.
| | | | | |
- | - | - | - | - |
$\blueE{1}$| $\tealE{2}$ |$\redE{4}$ |$ \redE{5}$ | $\tealE{10}$ | $\blueE{20}$
The next counting number is $5$. Since $5$ already appears on the list, we now have found all of the factor pairs for $20$.
[[☃ graded-group 1]]
[[☃ explanation 1]]
[[☃ explanation 2]]
##Multiples
Multiples are numbers that result when we multiply one whole number by another whole number. The first four multiples of $\blue{3}$ are $3, 6, 9$, and $12$ because:
$\blue{3} \times 1 = 3$
$\blue{3} \times 2 = 6$
$\blue{3} \times 3 = 9$
$\blue{3} \times 4 = 12$
Some other multiples of $\blue{3}$ are $15, 30$ and $300$.
$\blue{3} \times 5 = 15$
$\blue{3} \times 10 = 30$
$\blue{3} \times 100 = 300$
We can never list all of the multiples of a number. In our example, $3$ could be multiplied by an infinite number of numbers to find new multiples.
###Practice problems
The first multiple of any number is the number itself.
$7 \times 1 = 7$.
[[☃ graded-group 5]]
[[☃ explanation 5]]
The list shows multiples of $4$.
$4, 8, 12, 16, ...$
[[☃ graded-group 6]]
[[☃ explanation 6]]
The list shows multiples of $8$.
[[☃ graded-group 4]]
[[☃ explanation 4]]
[[☃ graded-group 1]]
[[☃ explanation 1]]
###Picturing multiples
The following pictures show multiples of $4$.
$4 \times 1 = 4$
[[☃ image 1]]
$4 \times 2 = 8$
[[☃ image 2]]
$4 \times 3 = 12$
[[☃ image 3]]
The next box will include the next multiple of $4$.
[[☃ graded-group 3]]
[[☃ explanation 3]]
###How do factors and multiples relate?
$\goldD{4}$ and $\greenD{7}$ are both** factors** of $\blueD{28}$ because they both divide evenly into $\blueD{28}$.
$\blueD{28}$ is a multiple of $\goldD{4},$ and it is also a multiple of $\greenD{7}$.
[[☃ graded-group 2]]
[[☃ explanation 2]]
##Practice with factors and multiples
[[☃ graded-group 1]]
[[☃ explanation 1]]
[[☃ graded-group 5]]
[[☃ explanation 5]]
We know that $9 \times 6 = 54$
[[☃ graded-group 4]]
[[☃ explanation 4]]
##Factors and multiples challenge
Factors and multiples are used when solving problems about the side lengths and areas of rectangles.
[[☃ explanation 3]]
A rectangle has an area of $50$ square cm.
[[☃ graded-group 1]]
[[☃ explanation 1]]
Mr. Trimble is putting out $36$ chocolate chip cookies for the students in his art club.
[[☃ graded-group 2]]
[[☃ explanation 2]]