Factor

A factor is a whole number that divides into a larger number without a remainder.

A factor is any of the two or more whole numbers we multiply to get a product. For example: 6×2=12. The numbers 6 and 2 are factors of 12.

A factor is also described as a whole number that divides into a larger number without a remainder. For example: $\frac{12}{6}$=2. The number 6 divides perfectly into 12. Therefore, the number 6 is a factor of 12.

Finding the factors of a number

1. List 1 and the number

Any number larger than 1 has the number itself and 1 as factors

2. Divide with 2

Start with 2 and see if it divides the number without a remainder.

If it does, write down 2 and the number of times 2 divides that number

3. Repeat step 2 with different numbers

Try dividing with consecutive numbers, 3,4,5,6,...

When you are able to divide without a remainder, write down that number and the number of times it was able to divide the number whose factor you are finding.

You stop when you get to the number which has already been listed as a factor in the "Second" column.

Examples

List the factors of following numbers:

1. 36
2. 12
3. 30
4. 60

Solution

1. Factors of 36

Step 1

Write 1 and the number whose factors you are finding.

Step 2

Try 2 and consecutive numbers to see if they divide the number without a remainder and write that number and the number of times it divides the number whose factor you are seeking.

You stop when you get to the number which has already been listed as a factor in the "Second" column.

List all the numbers that where able to divide the number whose factors you seek.

First	Feedback	Second
1	Factor	36
2	Factor	18
3	Factor	12
4	Factor	9
5	Not
6	Factor	6

List the first and second numbers which were factors from the lowest to highest.

Factors of 36 = {1,2,3,4,6,9,12,18,36}

2. Factors of 12

First	Feedback	Second
1	Factor	12
2	Factor	6
3	Factor	4

Factors of 12 = {1,2,3,4,6,12}

3. Factors of 30

First	Feedback	Second
1	Factor	30
2	Factor	15
3	Factor	10
4	Not Factor
5	Factor	6

Factors of 30 = {1,2,3,5,6,10,15,30}

4. Factors of 60

First	Feedback	Second
1	Factor	60
2	Factor	30
3	Factor	20
4	Factor	15
5	Factor	12
6	Factor	10
7	Not factor
8	Not factor
9	Not factor

Prime factors

A prime number is a number that has only two factors: 1 and the number itself.

Any non-prime number can be broken down into prime factors. We can then write the number as a product of its prime factors.

The prime numbers between 1 and 100 are shaded on the board below

Finding the prime factors of a number

To find the prime factors of a number, follow the steps below:

1. Divide the number with the smallest prime factor until it cannot divide any further

2. Move to the next prime factor. Keep on dividing until it cannot divide any further.

3. Repeat step 2 until you get 1 as a remainder.

4. Write down the product (multiplication) of each of the prime factors in the division.

5. Simplify the indices using the multiplication law of indices.

Example 1

Find the prime factors 180

$$\begin{array}{cc}2& 180\\ 2& 90\\ 3& 45\\ 3& 15\\ 5& 5\\ & 1\end{array}$$

Prime factors of 180 = 2 x 2 x 3 x 3 x 5 = 2² x 3² x 5

Example 2

Find the prime factors of 60

$$\begin{array}{cc}2& 60\\ 2& 30\\ 3& 15\\ 5& 5\\ & 1\end{array}$$

Prime factors of 60 = 2 x 2 x 3 x 5 = 2² x 3 x 5

Example 3

Find the prime factors of 16

$$\begin{array}{cc}2& 16\\ 2& 8\\ 2& 4\\ 2& 2\\ & 1\end{array}$$

Prime factors of 16 = 2 x 2 x 2 x 2 = 2⁴

Common Factors

Two numbers have a common factor if the same number is a factor of both numbers.

Finding the common factors

1. List the factors of each of the numbers you are finding their common factors.

2. List the numers which appeared in each of the listed factors.

Example 1

Find the common factors of 10 and 30.

Solution

Factors of 10

First	Feedback	Second
1	Factor	10
2	Factor	5
3	Not factor
4	Not factor

Factors of 10 = {1,2,5,10}

Factors of 30

First	Feedback	Second
1	Factor	30
2	Factor	15
3	Factor	10
4	Not factor
5	Factor	6

Factors of 30 = {1,2,3,5,6,10,15,30}

1,2 and 10 can be found in both the factors of 10 and 30, hence are the common factors.

Common factors of 10 and 30 = {1,2,10}

Highest Common Factor

The highest common factor is the largest whole number that divides into two or more numbers without a remainder.

The highest common factor (H.C.F) of two numbers is the largest number that is a factor of both.

From our list of common factors of 10 and 30, you will realize, 10 is the largest in that least. Hence the highest common factor of 10 and 30 is 10.

Using prime factors to find H.C.F

1. List the prime factors of each of the numbers you are finding the highest common factor.

2. Multiply the prime factors which appear in both list of prime factors, selecting the ones with the lowest power

Example 1

Use the prime factors to find the highest common factor of 10 and 30.

Prime factors of 10

$$\begin{array}{cc}2& 10\\ 5& 5\\ & 1\end{array}$$

Prime factors of 10 = 2 x 5

Prime factors of 30

$$\begin{array}{cc}2& 30\\ 3& 15\\ 5& 5\\ & 1\end{array}$$

Prime factors of 30 = 2 x 3 x 5

Pick the prime factors with the lowest power which is found in both

H.C.F = 2 x 5 = 10

Example 2

Use the prime factors to find the H.C.F of 60 and 80

Prime factors of 60

$$\begin{array}{cc}2& 60\\ 2& 30\\ 3& 15\\ 5& 5\\ & 1\end{array}$$

Prime factors = 2 x 2 x 3 x 5 = 2² x 3 x 5

Prime factors of 80

$$\begin{array}{cc}2& 80\\ 2& 40\\ 2& 20\\ 2& 10\\ 5& 5\\ & 1\end{array}$$

Prime factors = 2 x 2 x 2 x 2 x 5 = 2⁴ x 5

The common prime factors are 2 and 5

The lowest power of 2 is 2, hence the prime factor for the H.C.F will be 2²

5 has the same power (1) which is not written.

The H.C.F of 60 and 80 = 2² x 5 = 4 x 5 = 20

Multiples

A multiple is a whole number into which a smaller number divides without a remainder.

The multiples of a given number are the numbers you get when you multiply the given number with the natural numbers.

Multiples are obtained by repeated adding the same number.For instance, the multiples of 2 are obtained as follows:

1 x 2 = 0+2 = 2
2 x 2 = 2+2 = 4
3 x 2 = 4+2 = 6
4 x 2 = 6+2 = 8

You can complete the rest. In most school these are memorized. During examination, you can use this approach to get the multiples of a number you might be having challenge recalling.

List the multiples of 12

1 x 12 = 0+12 = 12
2 x 12 = 12+12 = 24
3 x 12 = 24+12 = 36
4 x 12 = 36+12 = 48
5 x 12 = 48+12 = 60

You can list the rest.

Multiplying by a number with zeros at the end

When you multiply by a number with zeros at the end, the result is simply the multiplication of the number being multiplied and the digits before the zeros at the end and then joined by the number of zeros at the end.

For instance when you multiply a number by 10,100,1000,10000,...,1n0s, the result is simply the multiplication of the number by 1 and joining the number of zeros (n0s) to the result (the number x 1).

Where n0s means number of zeros.

When you multiply a number by 20,200,2000,20000,...,2n0s, the result is simply the multiplication of the number by 2 and joining the number of zeros (n0s) to the result (the number x 2).

For instance:

2 x 10 = (2 x 1) joined by 0 = 20
3 x 120 = (3 x 12) joined by 0 = 360
4 x 50 = (4 x 5) joined by 0 = 400
2 x 10400 = (2 x 104) joined by 00 = 20800
3 x 600 = (3 x 6) joined by 00 = 1800
4 x 100 = (4 x 1) joined by 00 = 400
20 x 11000000 = (2 x 11 joined by 0) joined by 000000 = 220000000

Common multiples

A common multiple is a multiple of two or more numbers.

A multiple is a number that can be divided by another number without a remainder. Two numbers have a common multiple if the same number is a multiple of both those numbers.

Finding the common multiples of numbers

1. List the multiples of the numbers

2. List the numbers found in both list

Example 1

Find two common multiples of 5 and 6.

Solution

Multiples of 5 = {5,10,15,20,25,30,35,40,45,50,55,60,...}
Multiples of 6 = {6,12,18,24,30,36,42,48,54,60,66,...}

Common multiples = {30,60,...}

Least Common Multiples

The least/lowest common multiple (L.C.M) is the smallest whole number into which two or more numbers divide without a remainder.

From the common multiples of 5 and 6 listed above, you will realize that the lowest common multiple is 30.

Using prime factors to find the L.C.M

1. List the prime factors of the numbers.

2. L.C.M is the product (multiplication) of each of the prime factors with their highest power.

Example 1

Use prime factors to find the L.C.M of 5 and 6.

Prime factor of 5

Prime factor of 5 is 5

5 has only two factors, 1 and itself. 5 is the only prime number in these factors.

NOTE:1 is not a prime number

Prime factors of 6

$$\begin{array}{cc}2& 6\\ 3& 3\\ & 1\end{array}$$

Prime factors of 6 = 2 x 3

The prime factors for both are 2,3 and 5

The L.C.M = 2 x 3 x 5 = 30

Example 2

Use prime factors to find the L.C.M of 60 and 45.

Prime factors of 60

$$\begin{array}{cc}2& 60\\ 2& 30\\ 3& 15\\ 5& 5\\ & 1\end{array}$$

Prime factors = 2 x 2 x 3 x 5 = 2² x 3 x 5

Prime factors of 45

$$\begin{array}{cc}3& 45\\ 3& 15\\ 5& 5\\ & 1\end{array}$$

Prime factors = 3 x 3 x 5 = 3² x 5

The L.C.M is product of each of the prime factors in their highest power

L.C.M of 60 and 45 = 2² x 3² x 5 = 4 x 9 x 5 = 180

Example 3

Find the L.C.M of 12 and 18 using prime factors.

Prime factors of 12

$$\begin{array}{cc}2& 12\\ 2& 6\\ 3& 3\\ & 1\end{array}$$

Prime factors = 2 x 2 x 3 = 2² x 3

Prime factors of 18

$$\begin{array}{cc}2& 18\\ 3& 9\\ 3& 3\\ & 1\end{array}$$

Prime factors = 2 x 3 x 3 = 2 x 3²

The prime factors are 2 and 3.

Find their products in their highest power

The L.C.M = 2² x 3² = 4 x 9 = 36

Differences between H.C.F and L.C.M

The summary of H.C.F and L.C.M is listed below. Carefully take not so you won't confuse H.C.F and L.C.M

Highest Common Factor (H.C.F)	Lowest Common Multiple (L.C.M)
A factor of two or more numbers	A multiple of two of more numbers
H.C.F must divide into the given numbers	Given numbers must divide into L.C.M
Only common prime factors to lowest powers	All prime factors to highest powers
Smaller than or equal to given numbers	Larger than or equal to given numbers

APPLICATION OF KNOWLEDGE

Question 1

Find the least common multiple multiple of 7,14 and 18.

A. 71418 B. 1764 C. 252 D. 126

[B.E.C.E 1990 Section A Question 23]

Solution

Question 2

Find the G.C.F (H.C.F) of 2³ x 3² and 2³ x 3⁴

A. 8 B. 9 C. 72 D. 648

[B.E.C.E 1991 Section A Question 9]

Solution

Question 3

Find the least whole number which must be added to 207 to make it divisible by 17.

A. 0 B. 3 C. 13 D. 14

[B.E.C.E 1992 Section A Question 6]

Solution

Question 4

Find the Highest Common Factor (HCF) of 18, 36 and 120.

A. 2³ x 3² x 5 B. 2³ x 3 x 5 C. 2 x 3 D. 3² x 2²

[B.E.C.E 1993 Section A Question 6]

Solution

Question 5

Which of the following is the set of factors of 12?

A. {12,6,4,3,2,1} B. {12,6,4,3,2} C. {12,6,4,2} D. {6,4,2,1}

[B.E.C.E 1994 Section A Question 1]

Solution

Question 6

What is the H.C.F of 48, 30 and 18?

A. 2 B. 3 C. 5 D. 6

[B.E.C.E 1995 Section A Question 3]

Solution

Question 7

Find the L.C.M of 18, 42 and 90.

A. 2 x 3² B. 2 x 3 x 7 C. 2 x 3²x 5 D. 2 x 3² x 5 x 7

[B.E.C.E 1997 Section A Question 9]

Solution

Question 8

What is the H.C.F of 18, 42 and 90?

A. 21 B. 18 C. 9 D. 6

[B.E.C.E 1997 Section A Question 10]

Solution

Question 9

Find the Least Common Multiple (L.C.M) of the following numbers: 3, 5 and 9.

[B.E.C.E 2019 Section B Question 1(b)]

Solution

Question 10

Given that P = {factors of 36} and Q = {factors of 54},

[B.E.C.E 2022 Section B Question 1(a)]

Solution