Integers and Number Relationships
3. Factors, Multiples, and Divisibility
Learning outcomes
- I can identify factors of a number.
- I can identify multiples of a number.
- I can use divisibility rules.
- I can determine whether one number is a factor of another.
- I can solve problems involving factors and multiples.
What Are Factors and Multiples?
Factors and multiples describe relationships between whole numbers.
Consider:
4 × 6 = 24
From this multiplication fact, we know:
- 4 is a factor of 24.
- 6 is a factor of 24.
- 24 is a multiple of 4.
- 24 is a multiple of 6.
Factors and multiples are closely connected through multiplication and division.
What Is a Factor?
A factor of a whole number divides that number exactly, leaving no remainder.
For example:
3 is a factor of 12
because:
12 ÷ 3 = 4
with no remainder.
Similarly:
4 is a factor of 12
because:
12 ÷ 4 = 3
Factor Pairs
Factors often come in pairs.
For 12:
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12
Therefore, the factors of 12 are:
1, 2, 3, 4, 6, 12
Each multiplication equation gives us a factor pair.
Finding Factors Systematically
Suppose we want all the factors of:
24
Start with 1 and test possible divisors.
1 × 24 = 24
2 × 12 = 24
3 × 8 = 24
4 × 6 = 24
Therefore:
Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
A systematic approach helps make sure no factors are missed.
Factors Using Division
We can also test factors using division.
Is 5 a factor of 35?
Calculate:
35 ÷ 5 = 7
There is no remainder.
Therefore:
5 is a factor of 35.
When a Number Is Not a Factor
Is 4 a factor of 18?
Calculate:
18 ÷ 4 = 4 remainder 2
Since the division does not produce a whole-number quotient:
4 is not a factor of 18.
Every Number Has Certain Factors
Every positive whole number has at least:
1 and itself
as factors.
For example:
Factors of 17:
1, 17
Factors of 20 include:
1 and 20
Factors of 100 include:
1 and 100
The number 1 is a factor of every positive whole number.
What Is a Multiple?
A multiple is produced when a number is multiplied by a whole number.
Multiples of 5 include:
5, 10, 15, 20, 25, 30, 35, ...
because:
5 × 1 = 5
5 × 2 = 10
5 × 3 = 15
5 × 4 = 20
and so on.
Multiples Continue Forever
A number has a limited number of positive factors, but it has infinitely many positive multiples.
For example, multiples of 7 include:
7, 14, 21, 28, 35, 42, 49, 56, ...
There is always another multiple because we can continue multiplying by larger whole numbers.
Multiples on a Number Line
Multiples can be visualized using equal jumps on a number line.
For multiples of 4:
4, 8, 12, 16, 20, 24, ...
Each jump has a size of:
4
This connects multiples with repeated addition.
Factors and Multiples Are Related
Consider:
6 × 8 = 48
This tells us:
6 is a factor of 48
8 is a factor of 48
and:
48 is a multiple of 6
48 is a multiple of 8
The relationship works in both directions.
Factor or Multiple?
Consider the numbers:
5 and 30
Since:
5 × 6 = 30
we can say:
5 is a factor of 30
and:
30 is a multiple of 5
The words describe different sides of the same relationship.
Factor Example
Is:
9
a factor of:
72?
Calculate:
72 ÷ 9 = 8
Since the quotient is a whole number:
Yes, 9 is a factor of 72.
Multiple Example
Is:
72
a multiple of:
9?
Since:
9 × 8 = 72
the answer is:
Yes, 72 is a multiple of 9.
What Does Divisible Mean?
A number is divisible by another number if the division produces a whole-number quotient with no remainder.
For example:
24 is divisible by 6
because:
24 ÷ 6 = 4
But:
25 is not divisible by 6
because the division leaves a remainder.
Divisibility and Factors
If:
a number is divisible by another number
then that second number is a factor.
For example:
42 is divisible by 7
so:
7 is a factor of 42
Divisibility Rules
Divisibility rules are shortcuts that help determine whether one number divides another exactly.
Instead of performing long division every time, we can examine the digits of the number.
Useful divisibility rules include rules for:
2, 3, 4, 5, 6, 8, 9, and 10
Divisibility Rule for 2
A number is divisible by 2 if its last digit is:
0, 2, 4, 6, or 8
These are even digits.
Examples:
38
ends in 8, so:
38 is divisible by 2
124
ends in 4, so:
124 is divisible by 2
Divisibility Rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
Consider:
123
Add the digits:
1 + 2 + 3 = 6
Since 6 is divisible by 3:
123 is divisible by 3
Another Rule of 3 Example
Is:
742
divisible by 3?
Add:
7 + 4 + 2 = 13
13 is not divisible by 3.
Therefore:
742 is not divisible by 3
Divisibility Rule for 4
A number is divisible by 4 if the number formed by its final two digits is divisible by 4.
Consider:
316
Look at:
16
Since:
16 ÷ 4 = 4
we know:
316 is divisible by 4
Another Rule of 4 Example
Consider:
742
Look at the final two digits:
42
42 is not divisible by 4.
Therefore:
742 is not divisible by 4
Divisibility Rule for 5
A number is divisible by 5 if its last digit is:
0 or 5
Examples:
35
120
1,005
are all divisible by 5.
But:
127
is not divisible by 5.
Divisibility Rule for 6
A number is divisible by 6 if it is divisible by:
2 and 3
Both conditions must be true.
Consider:
126
It is even, so it is divisible by 2.
Digit sum:
1 + 2 + 6 = 9
9 is divisible by 3.
Therefore:
126 is divisible by 6
Rule of 6: A Common Mistake
Consider:
27
The digit sum is:
2 + 7 = 9
So 27 is divisible by 3.
However, 27 is not even.
Therefore:
27 is not divisible by 6
Both the 2 rule and the 3 rule must work.
Divisibility Rule for 8
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Consider:
2,136
Look at:
136
Since:
136 ÷ 8 = 17
we know:
2,136 is divisible by 8
Divisibility Rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Consider:
729
Digit sum:
7 + 2 + 9 = 18
18 is divisible by 9.
Therefore:
729 is divisible by 9
Divisibility Rule for 10
A number is divisible by 10 if its final digit is:
0
Examples:
40
120
2,350
are divisible by 10.
Numbers such as:
45
126
are not divisible by 10.
Useful Divisibility Rules
- 2: Last digit is 0, 2, 4, 6, or 8.
- 3: Sum of digits is divisible by 3.
- 4: Last two digits form a number divisible by 4.
- 5: Last digit is 0 or 5.
- 6: Number is divisible by both 2 and 3.
- 8: Last three digits form a number divisible by 8.
- 9: Sum of digits is divisible by 9.
- 10: Last digit is 0.
These rules make factor testing much faster.
Using More Than One Divisibility Rule
Consider:
360
Is it divisible by 2?
Yes. It ends in 0.
By 3?
3 + 6 + 0 = 9
Yes.
By 4?
Last two digits:
60
Yes.
By 5?
Yes. It ends in 0.
By 6?
Yes. It is divisible by both 2 and 3.
By 8?
360 ÷ 8 = 45
Yes.
By 9?
Digit sum is 9.
Yes.
By 10?
Yes.
This tells us many possible factors of 360 without having to test each one using long division.
Factor Trees
Factors can also be represented using a factor tree.
Consider:
24
We could begin:
24 = 4 × 6
Then:
4 = 2 × 2
and:
6 = 2 × 3
So:
24 = 2 × 2 × 2 × 3
or:
24 = 2³ × 3
This is called prime factorization.
Prime Numbers
A prime number has exactly two positive factors:
1 and itself
Examples:
2, 3, 5, 7, 11, 13, 17, 19
For example, the only factors of 7 are:
1 and 7
Therefore:
7 is prime
Composite Numbers
A composite number has more than two positive factors.
For example:
Factors of 12:
1, 2, 3, 4, 6, 12
Therefore:
12 is composite
The number:
1
is neither prime nor composite because it has only one positive factor.
Worked Example 1: Find All Factors
Find the factors of:
18
Factor pairs:
1 × 18
2 × 9
3 × 6
Therefore:
Factors of 18 = 1, 2, 3, 6, 9, 18
Worked Example 2: Find Multiples
Write the first six positive multiples of 8.
Calculate:
8 × 1 = 8
8 × 2 = 16
8 × 3 = 24
8 × 4 = 32
8 × 5 = 40
8 × 6 = 48
Answer:
8, 16, 24, 32, 40, 48
Worked Example 3: Test a Factor
Is 7 a factor of 91?
Calculate:
91 ÷ 7 = 13
There is no remainder.
Therefore:
Yes, 7 is a factor of 91.
Worked Example 4: Use a Divisibility Rule
Is:
438
divisible by 3?
Add the digits:
4 + 3 + 8 = 15
15 is divisible by 3.
Therefore:
438 is divisible by 3.
Worked Example 5: Divisibility by 6
Is:
234
divisible by 6?
First test 2:
234 is even.
So yes.
Test 3:
2 + 3 + 4 = 9
9 is divisible by 3.
Therefore:
234 is divisible by 6.
Common Factors
Two or more numbers can share factors.
Consider:
12 and 18
Factors of 12:
1, 2, 3, 4, 6, 12
Factors of 18:
1, 2, 3, 6, 9, 18
Common factors:
1, 2, 3, 6
The largest common factor is:
6
This is called the greatest common factor (GCF).
Greatest Common Factor
The greatest common factor is the largest factor shared by two or more numbers.
For example:
Factors of 20:
1, 2, 4, 5, 10, 20
Factors of 30:
1, 2, 3, 5, 6, 10, 15, 30
Common factors:
1, 2, 5, 10
Therefore:
GCF(20, 30) = 10
Common Multiples
Two numbers can also share multiples.
Consider:
Multiples of 4:
4, 8, 12, 16, 20, 24, 28, ...
Multiples of 6:
6, 12, 18, 24, 30, 36, ...
Common multiples include:
12, 24, 36, ...
The smallest positive common multiple is:
12
This is called the least common multiple (LCM).
Least Common Multiple
The least common multiple is the smallest positive multiple shared by two or more numbers.
For example:
Multiples of 5:
5, 10, 15, 20, 25, 30, ...
Multiples of 6:
6, 12, 18, 24, 30, ...
The first common multiple is:
30
Therefore:
LCM(5, 6) = 30
Factors in Equal Grouping Problems
Suppose:
24 students
must be placed into equal groups with no students left over.
Possible group sizes are factors of 24:
1, 2, 3, 4, 6, 8, 12, 24
For example:
24 ÷ 6 = 4
so groups of 6 are possible.
But groups of 5 are not possible because:
24 ÷ 5
leaves a remainder.
Real-World Problem 1: Arranging Objects
A teacher has:
36 chairs
and wants to arrange them into equal rows.
Possible row sizes must be factors of 36.
Factor pairs:
1 × 36
2 × 18
3 × 12
4 × 9
6 × 6
Therefore, possible numbers of chairs per row include:
1, 2, 3, 4, 6, 9, 12, 18, 36
Real-World Problem 2: Packaging
A company has:
48 bottles
and wants to place the same number in each box with none left over.
Could each box contain:
6 bottles?
Calculate:
48 ÷ 6 = 8
Yes.
Therefore:
6 is a factor of 48
and 48 bottles can be packed into:
8 boxes of 6
Real-World Problem 3: Repeating Events
One bus arrives every:
10 minutes
Another arrives every:
15 minutes
If they arrive together now, when will they next arrive together?
Multiples of 10:
10, 20, 30, 40, ...
Multiples of 15:
15, 30, 45, ...
The least common multiple is:
30
Therefore:
They will next arrive together in 30 minutes.
Real-World Problem 4: Making Equal Packs
A teacher has:
24 pencils
and:
36 erasers
and wants to make the greatest possible number of identical packs without leftovers.
We need the GCF.
Factors shared by 24 and 36 include:
1, 2, 3, 4, 6, 12
The greatest is:
12
Therefore:
12 identical packs
can be made.
Each pack contains:
24 ÷ 12 = 2 pencils
and:
36 ÷ 12 = 3 erasers
Real-World Problem 5: Flashing Lights
A red light flashes every:
4 seconds
A blue light flashes every:
6 seconds
They flash together now.
When will they next flash together?
Multiples of 4:
4, 8, 12, 16, ...
Multiples of 6:
6, 12, 18, ...
LCM:
12
Therefore:
They will flash together again after 12 seconds.
Recognizing Factor Problems
A problem may involve factors when it asks about:
- equal groups
- equal rows
- arranging objects
- sharing without leftovers
- possible dimensions
- dividing quantities exactly
- largest identical groups
Questions involving the greatest possible number of equal groups often involve the GCF.
Recognizing Multiple Problems
A problem may involve multiples when it asks about:
- repeating patterns
- schedules
- cycles
- events occurring together
- skip counting
- future times when patterns match
Questions asking when two repeating events will next occur together often involve the LCM.
Factor or Multiple? Example
Consider:
8 and 56
Since:
8 × 7 = 56
we know:
8 is a factor of 56
and:
56 is a multiple of 8
We can also say:
56 is divisible by 8
All three statements describe the same mathematical relationship.
Using Divisibility Rules to Find Factors
Suppose we want to know whether 3 is a factor of:
2,451
Instead of dividing, add the digits:
2 + 4 + 5 + 1 = 12
12 is divisible by 3.
Therefore:
2,451 is divisible by 3
and:
3 is a factor of 2,451
Divisibility rules are particularly useful with large numbers.
Challenge Example
Determine whether:
7,236
is divisible by:
2, 3, 4, 5, 6, 9, and 10
By 2:
Last digit is 6.
Yes
By 3:
7 + 2 + 3 + 6 = 18
Yes
By 4:
Last two digits are 36.
Yes
By 5:
Does not end in 0 or 5.
No
By 6:
Divisible by both 2 and 3.
Yes
By 9:
Digit sum is 18.
Yes
By 10:
Does not end in 0.
No
A Reliable Strategy for Finding Factors
Step 1: Begin with 1.
Step 2: Test whether it divides the number exactly.
Step 3: Record both numbers in the factor pair.
Step 4: Continue testing larger numbers.
Step 5: Stop once the factor pairs begin repeating.
Step 6: List the factors in order.
For:
30
factor pairs are:
1 × 30
2 × 15
3 × 10
5 × 6
So:
Factors = 1, 2, 3, 5, 6, 10, 15, 30
A Reliable Strategy for Finding Multiples
To find multiples:
Step 1: Choose the number.
Step 2: Multiply it by 1, 2, 3, 4, and so on.
For 9:
9 × 1 = 9
9 × 2 = 18
9 × 3 = 27
9 × 4 = 36
9 × 5 = 45
Therefore:
9, 18, 27, 36, 45, ...
Common Mistakes
Mistake 1: Confusing factors and multiples
For:
4 and 20
4 is a factor of 20.
20 is a multiple of 4.
Mistake 2: Forgetting 1 and the number itself
The factors of 10 are:
1, 2, 5, 10
not just:
2 and 5
Mistake 3: Stopping a multiples list
Multiples continue forever.
There is no largest multiple of a positive whole number.
Mistake 4: Thinking 1 is prime
1 has only one positive factor.
A prime number must have exactly two.
Therefore:
1 is neither prime nor composite.
Mistake 5: Using only one test for divisibility by 6
A number must be divisible by:
both 2 and 3
to be divisible by 6.
Mistake 6: Confusing GCF and LCM
GCF involves:
shared factors
LCM involves:
shared multiples
Error Analysis
A student says:
6 is a multiple of 24 because 6 × 4 = 24.
The multiplication fact is correct, but the relationship has been reversed.
Since:
6 × 4 = 24
we know:
6 is a factor of 24
and:
24 is a multiple of 6
Another Error Analysis
A student says:
123 is not divisible by 3 because it does not end in 3, 6, or 9.
This uses the wrong rule.
For divisibility by 3, add the digits:
1 + 2 + 3 = 6
Since 6 is divisible by 3:
123 is divisible by 3
In fact:
123 ÷ 3 = 41
Did You Know?
Divisibility rules work because of the structure of our base-ten number system.
For example, consider:
372
We can write it as:
300 + 70 + 2
For divisibility by 3:
300, 60, and 0
are all divisible by 3, so the remainder depends on the digit sum:
3 + 7 + 2 = 12
Since 12 is divisible by 3:
372 is divisible by 3
Divisibility rules are not simply tricks; they come from mathematical patterns in place value.
Key Terms
- Factor: Whole number that divides another whole number exactly.
- Factor pair: Two factors whose product equals a given number.
- Multiple: Result of multiplying a number by a whole number.
- Divisible: Able to be divided exactly without a remainder.
- Divisibility rule: Shortcut for determining whether a number is divisible by another number.
- Remainder: Amount left after division when division is not exact.
- Prime number: Whole number greater than 1 with exactly two positive factors.
- Composite number: Whole number greater than 1 with more than two positive factors.
- Prime factorization: Writing a number as a product of prime factors.
- Factor tree: Diagram used to break a number into factors.
- Common factor: Factor shared by two or more numbers.
- Greatest common factor (GCF): Largest factor shared by two or more numbers.
- Common multiple: Multiple shared by two or more numbers.
- Least common multiple (LCM): Smallest positive multiple shared by two or more numbers.
Key Relationships
If:
a × b = c
then:
a and b are factors of c
and:
c is a multiple of a and b
For example:
7 × 8 = 56
Therefore:
7 and 8 are factors of 56
and:
56 is a multiple of 7 and 8
Also:
56 ÷ 7 = 8
56 ÷ 8 = 7
so 56 is divisible by both 7 and 8.
Key Divisibility Rules
Divisible by 2: Last digit is 0, 2, 4, 6, or 8.
Divisible by 3: Sum of digits is divisible by 3.
Divisible by 4: Last two digits form a number divisible by 4.
Divisible by 5: Last digit is 0 or 5.
Divisible by 6: Divisible by both 2 and 3.
Divisible by 8: Last three digits form a number divisible by 8.
Divisible by 9: Sum of digits is divisible by 9.
Divisible by 10: Last digit is 0.
Key Takeaways
- A factor divides another whole number exactly without a remainder.
- Factors can be found using multiplication pairs or division.
- Every positive whole number has 1 and itself as factors.
- A multiple is produced by multiplying a number by a whole number.
- Positive multiples continue indefinitely.
- Factors and multiples describe opposite sides of the same multiplication relationship.
- If 5 is a factor of 30, then 30 is a multiple of 5.
- A number is divisible by another number when the quotient is a whole number with no remainder.
- Divisibility rules allow us to test numbers efficiently without performing full division.
- Divisibility by 2 depends on the final digit.
- Divisibility by 3 and 9 depends on the sum of the digits.
- Divisibility by 4 depends on the final two digits.
- Divisibility by 5 depends on whether the number ends in 0 or 5.
- Divisibility by 6 requires divisibility by both 2 and 3.
- Divisibility by 8 can be tested using the final three digits.
- Divisibility by 10 requires a final digit of 0.
- Prime numbers have exactly two positive factors.
- Composite numbers have more than two positive factors.
- The number 1 is neither prime nor composite.
- Common factors can be used to find the greatest common factor.
- Common multiples can be used to find the least common multiple.
- Factor problems often involve equal groups, arrangements, and sharing without leftovers.
- Multiple problems often involve repeating events, cycles, and schedules.
- Factors, multiples, and divisibility form an important foundation for fractions, ratios, prime factorization, algebra, and number theory.