Learn about factors and multiples and how they relate to each other.

Faktorer

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 1212.
1212 dots can be arranged in 1\tealC{1} row with 12\tealC{12} dots.
1×12=12\tealC{1 \times 12} = 12

1212 dots can also be arranged in 2\purpleC{2} rows with 6\purpleC{6} dots per row.
2×6=12\purpleC{2 \times 6} = 12
Or, we can arrange 1212 dots in 3\redC{3} rows with 4\redC{4} dots in each row.
3×4=12\redC{3 \times 4} = 12
Once we figure out all of the ways that 1212 dots can be arranged, we can look at the number of rows and number of dots in each row to find the factors of 1212.
1\tealC{1}, 12\tealC{12}, 2\purpleC{2}, 6\purpleC{6}, 3\redC{3}, and 4\redC{4} are all factors of 1212.
We can make 1212 with a row of 5\greenD5 and a row of 7\goldD7. So are 5\greenD5 and 7\goldD7 factors of 1212?

No. 5\greenD5 and 7\goldD7 are not factors because the dots are not divided into equal sized groups.

Finding factors without pictures

We can find the factors for 1616 without drawing dots by thinking about numbers that will divide into 1616 evenly.
1\blueD{1} is a factor of 1616 because 1\blueD{1} can be divided into 1616 with no remainder.
16÷1=1616 \div \blueD{1}= \blueD{16}
The quotient, which is 16,\blueD{16}, is also a factor of 1616.
2\greenD{2} is a factor of 1616 because 2\greenD{2} can be divided into 1616 with no remainder.
16÷2=816 \div \greenD{2}= \greenD{8}
The quotient, which is 8\greenD{8}, is also a factor of 1616.
4\purpleD{4} is a factor of 1616 because 4\purpleD{4} can be divided into 1616 with no remainder.
16÷4=416 \div \purpleD{4}= \purpleD{4}
In this case the quotient is 4\purpleD{4}, which we have already discovered is a factor of 1616.
The factors of 1616 are 1,16\blueD{1, 16}, 2,8,\greenD{2, 8}, and 4\purpleD{4}.
Numbers like 33 and 55 are not factors of 1616 because they cannot be divided evenly into 1616.

Factor hints

Every number has 11 as a factor.
11 is a factor of 1010.
11 is a factor of 364364.
11 is a factor of 5,7875,787.
Every number has itself as a factor.
4141 is a factor of 4141.
128128 is a factor of 128128.
4,3794,379 is a factor of 4,3794,379.

Factor pairs

Two numbers that we multiply together to get a certain product are called factor pairs. To get the product of 88, we can multiply 1\purpleD{1} ×\times 8\purpleD{ 8} and 2\greenD{2} ×\times 4\greenD{4}. So the factor pairs for 88 are 1\purpleD{1} and 8\purpleD{8} and 2\greenD{2} and 4\greenD{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.

Let's find the factor pairs of 2020. Remember, we are looking for two whole numbers that we can multiply together to get 2020.
We'll start with 1\blueD{1} because we know that 1\blueD{1} is a factor for every number. We multiply 1×20\blueD{1} \times \blueD{20}, to get 2020, so 20\blueD{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.
1\blueD{1}20\blueD{20}
Now we check to see whether the next counting number, 22, is a factor.
Is there a whole number we can multiply by 2\greenD{2} to get 2020? Yes. 2×10=20\greenD{2} \times \greenD{10}=20. So 2\greenD{2} and 10\greenD{10} are another factor pair.
1\blueD{1}2\greenD{2}10\greenD{10}20\blueD{20}
The next counting number is 33. Is there a whole number we can multiply by 33 to get 2020? No. So 33 is not a factor of 2020.
Can we multiply 4\redD{4} by a whole number to get 2020? Yes. 4×5=20\redD{4} \times \redD{5}= 20. So 4\redD{4} and 5\redD{5} are a factor pair.
1\blueD{1}2\greenD{2}4\redD{4}5 \redD{5}10\greenD{10}20\blueD{20}
The next counting number is 55. Since 55 already appears on the list, we now have found all of the factors pairs for 2020.

Multiples

Multiples are numbers that result when we multiply one whole number by another whole number. The first four multiples of 3\blue{3} are 3,6,93, 6, 9, and 1212 because:
3×1=3\blue{3} \times 1 = 3
3×2=6\blue{3} \times 2 = 6
3×3=9\blue{3} \times 3 = 9
3×4=12\blue{3} \times 4 = 12
Some other multiples of 3\blue{3} are 15,3015, 30 and 300300.
3×5=15\blue{3} \times 5 = 15
3×10=30\blue{3} \times 10 = 30
3×100=300\blue{3} \times 100 = 300
We can never list all of the multiples of a number. In our example, 33 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.
7×1=77 \times 1 = 7.
The list shows multiples of 44.
4,8,12,16,...4, 8, 12, 16, ...
The list shows multiples of 88.

Picturing multiples

The following pictures show multiples of 44.
4×1=44 \times 1 = 4
4×2=84 \times 2 = 8
4×3=124 \times 3 = 12
The next box will include the next multiple of 44.

How do factors and multiples relate?

4\goldD{4} and 7\greenD{7} are both factors of 28\blueD{28} because they both divide evenly into 28\blueD{28}.
28\blueD{28} is a multiple of 4,\goldD{4}, and it is also a multiple of 7\greenD{7}.

Practice with factors and multiples

We know that 9×6=549 \times 6 = 54

Factors and multiples challenge

Factors and multiples are used when solving problems about the side lengths and areas of rectangles.
A rectangle has an area of 5050 square cm.
Mr. Trimble is putting out 3636 chocolate chip cookies for the students in his art club.