### Factors and Divisibility

When two non-zero integers are multiplied, each integer is a factor of the product.

For example, 4 × 5 = 20, so 4 and 5 are factors of 20.

The integer a is said to be divisible by the integer b if b is a factor of a. This means a can be divided by b with an integer result (meaning there is no remainder).

For example, 20 is divisible by 4 because 4 is a factor of 20, so 20/4 = 5. On the other hand, 8 is not a factor of 20 because 20/8 = 2.5, and 2.5 is not an integer.

#### Rules of Divisibility

1. All integers are divisible by 1.
2. All integers with a units digit of 0, 2, 4, 6, or 8 are divisible by 2.
3. If the sum of an integer’s digits is divisible by 3, then the integer is divisible by 3. For example, 222 consists of digits that add to 6, so the integer is divisible by 3 (222 / 3 = 74).
4. An integer is divisible by 4 if the number formed by its last two digits is divisible by 4. For example, 6,248 is divisible by 4, but 6,250 is not.
5. An integer is divisible by 5 if it ends in 0 or 5.
6. An integer is divisible by 6 if it is even (divisible by 2) and also divisible by 3. Any even number divisible by 3 is also divisible by 6.
7. A number is divisible by 8 if it is divisible by 2 three times, or if the last three digits are divisible by 8. For example, 24 is divisible by 8: 24 ÷ 2 = 12, 12 ÷ 2 = 6, 6 ÷ 2 = 3. For larger numbers, just check the last three digits. For example, 11,728 is divisible by 8 because 728 is divisible by 8 (728 ÷ 8 = 91).
8. If the sum of the digits is divisible by 9, then the integer is divisible by 9.
For example: 1,044 has digits that add to 9 (1 + 0 + 4 + 4 = 9) so 1,044 is divisible by 9 (1,044/9 =116).
9. An integer is divisible by 10 if it ends in 0.

### Factoring

By factoring a number, you break it down into as many factors as possible. For small numbers, an easy way to do this is in a table. Use the rules of divisibility to check for possible factors.

1

12

2

6

3

4

#### Factors of 105

##### Large

1

105

3

35

5

21

7

15

The factors of 12 are 1, 2, 3, 4, 6, and 12.

The factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105.

Don’t forget!

When a question asks, “how many factors does ____ have?”
Always remember to include:

• 1, because 1 is a factor of ALL integers
• The number itself, because any integer is divisible by itself

For example, 21 = 1 × 21, so 1 and 21 are factors of 21.

##### Example

State all the factors of 10.

### Solution

The number 10 can be divided by 1, 2, 5, and 10. Hence, these are its factors.

##### Example

State all the factors of 63.

### Solution

The number 63 can be divided by 1, 3, 7, 9, 21, and 63. Hence, these are its factors.

### Prime Numbers

A prime number is an integer greater than 1 whose only positive factors are 1 and itself.

The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. You should memorize these common primes.

Note: The number 1 is not a prime number.  Except for 2, all prime numbers are odd numbers. This means that the sum of two prime numbers will be even, unless one of the prime numbers is 2.

### Prime Factorization

Using your knowledge of prime numbers, you can break a number down into its prime factors.

You do this by finding factors using the divisibility rules.  Start with the smallest factors.
140 = 2 × 70 = 2 × 2 × 35 = 2 × 2 × 5 × 7

Another way to see prime factorization is to use a factor tree. The tree shows the prime factors.  The prime factors of 72 are 2 × 2 × 2 × 3 × 3.

##### Example

Is 36 a factor of 162?

### Solution

Compare the prime factors. Any factor of a number is the product of elements of a subset of the number’s prime factors.

36 = 2 × 2 × 3 × 3
162 = 2 × 3 × 3 × 3 × 3

36 shares the factors 2 × 3 × 3 with 162, but 36 has an additional factor of 2 that 162 does not have.  So 36 is not a factor of 162.

##### Example

If the integer n is divisible by 3, 5 and 13, what other numbers must be factors of n?

### Solution

Since we know that 3, 5, and 13 are prime factors of n, we can deduce that n must be divisible by all the products of the primes.

So we need to find the products of any combination of 3, 5 and 13.

(3 × 5) = 15 must be a factor
(3 × 13) = 39 must be a factor
(5 × 13) = 65 must be a factor
(3 × 5 × 13) = 195 must be a factor

Use the handy prime factorization chart to visualize if you need to.

The factors of n include 1, 3, 5, 13, 15, 39, 65 and 195.

### The Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor (divisor) for two numbers. For example, the GCF of 10 and 15 is 5.

The primary use of GCF is to reduce or simplify fractions. (There are examples showing the GCF and fractions in the next section.)

How to get the GCF

• List the prime factors of each number.
• See which prime factors are in both lists
• Multiply the factors both numbers share. If there are no common prime factors, then the GCF is 1
##### Example

What is the GCF of 36 and 162?

### Solution

Compare the prime factors.

36 = 2 × 2 × 3 × 3                               162 = 2 × 3 × 3 × 3 × 3

36 and 162 share the factors 2 × 3 × 3, so the GCF is 2 × 3 × 3 = 18.

##### Example

If a and b are prime numbers greater than 5, then what is the GCF of 6ab and 10a2?

### Solution

Compare the prime factors.

6ab = 2 × 3 × a × b                             10a2 = 2 × 5 × a × a

These expressions share the factors 2 × a, so the GCF is 2a.

### Factors and Multiples

A multiple of an integer n is the product of n and any integer. Sometimes it is easy to confuse factors and multiples.

• Factors divide an integer, so are less than or equal to the integer. An integer has only a limited number of factors.
• Multiples are greater than or equal to the integer. There are an infinite number of multiples of an integer.

For example, the factors of 12 are 1, 2, 3, 4, 6 and 12.  The first 10 multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108 and 120.

Factors, multiples, and divisibility are closely related.  For example, these three sentences say the same thing.

• 3 is a factor of 12.
• 12 is a multiple of 3.
• 12 is divisible by 3.

Using variables: x, y, and n are integers, and y = nx. So:

• x is a factor of y
• y = nx is a multiple of x
• y is divisible by x

### Least Common Multiple (LCM)

A common multiple of two or more integers is an integer that is a multiple of all of them. You can find the least common multiple by making a list of multiples for all the integers and identifying the smallest number that shows up on all the lists – this number will be the LCM.

The primary use of LCM is to add or subtract fractions. (There are examples showing the LCM and fractions in the next section.)

For example, these are multiples of 4:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …

Multiples of 6:
6, 12, 18, 24, 30, 36, 42, 48, …

The common multiples shown in the lists are 12, 24 and 36.

The LCM is 12.

### GCF, LCM, and Prime Factors

Another way to see the GCF and LCM is to use a Venn diagram and the prime factors.

The GCF is the product of the shared factors.

The LCM is the product of all the factors, without repeating the shared factors.

For 120 and 132, the Venn diagram shows:
GCF  2 × 2 × 3 = 12
LCM   2 × 5 × (2 × 2 × 3) × 11 = 1320

##### Example

What is the LCM of 90 and 108?

### Solution

Compare the prime factors.

90 = 2 × 3 × 3 × 5                               108 = 2 × 2 × 3 × 3 × 3

90 and 108 share the factors 2 × 3 × 3.  90 also has the factor 5, and 108 has the additional factors 2 × 3.

So the LCM is 5 × (2 × 3 × 3× 2 × 3 = 540.

##### Example

If the GCF of 2a and 7b is 1, then what is the LCM of 12ab and 42b2?

### Solution

Let’s factor first:

12ab = 2 × 2 × 3 × a × b                                 42b2 = 2 × 3 × 7 × b × b

Note that the factors a and b are not necessarily prime.

Then we group the common factors:

12ab = (2 × 3 × b) × 2a                                 42b2 = (2 × 3 × b) × 7b

Since we know that the GCF of 2a and 7b is 1, they share no prime factors. These expressions share (2 × 3 × b) only, so the LCM is

× a × (2 × 3 × b× 7 × b = 84ab2.

##### Example

How many unique prime factors of 2,100 are there?

### Solution

Do the prime factorization.

2,100 = 2 × 2 × 3 × 5 × 5 × 7

So there are 4 different prime factors of 2,100: 2, 3, 5, and 7.

Because of the word “unique” in this question, do NOT count the repeating prime factors.

##### Example

If x = 40y, then what is the greatest common factor of x and 18y, in terms of y?

### Solution

Since x = 40y, the question is asking us to find the GCF of 40y and 18y.

One common factor is y.

Factor 40 and 18 to find the common prime factors.

18 = 3 × 3 × 2
40 = 2 × 2 × 2 × 5

The greatest common factor of 18 and 40 is 2. Combining, the desired GCF is 2y.

In a question like this, especially with larger numbers, you could also use the methods Backsolve (using the answer choices) or Plug-In (trying numbers to test that 2y is the correct answer).

##### Example

If x is a positive even integer, then what are the GCF and LCM of the expressions x2 + 5x + 6 and x2 + 3x + 2?

### Solution

Just as with numbers, you need to compare the factors.

x2 + 5x + 6 = (x + 2)(x + 3)

x2 + 3x + 2 = (x + 2)(x + 1)

(x + 2) is a common factor, while (x + 3) and (x + 1) are two consecutive odd integers. The GCF of two consecutive odd integers is 1, because if we divide (x + 3) by any factor of (x + 1), the remainder will be 2.

Therefore the GCF of x2 + 5x + 6 and x2 + 3x + 2 is (x + 2).

The least common multiple will be the product (x + 2)(x + 3)(x + 1).  The answer choices may leave the expressions as factors, or may multiply the factors:

x3 + 6x2 + 11x + 6

To see further discussion of this question in our forum, click here.

Factoring quadratic expressions will be covered in the Algebra section.

##### Example

If the least common multiple of n and 12 is 36, what are the possible values of n?

### Solution

The largest possible value of n is 36.  An LCM is at least as large as any of the integers.

Now let’s look for other possible values of n.

Look at the prime factors of 12 and 36.

12 = 2 × 2 × 3

36 = 2 × 2 × 3 × 3

The LCM of n and 12 will contain the common factors 2 × 2 × 3 × 3. Can n have more factors? No.

The LCM contains 3 × 3, and 12 only has one factor of 3.  So n must have 2 factors of 3, or 3 × 3.

The LCM contains 2 × 2, and 12 provides them. So n can have 0, 1, or 2 factors of 2, and there will still be the 2 × 2 in the LCM.

So n contains exactly two factors of 3, and can contain 0, 1, or 2 factors of 2.  The three possible values of n are 3 × 3 = 9, 3 × 3 × 2 = 18, and 3 × 3 × 2 × 2 = 36.

To see further discussion of this question in our forum, click here.