 # Factors and Multiples – Differences & Examples By definition, a factor is a number that divides another number without leaving any remainder. In contrast, a multiple is a number that, when divided a certain of times by another number, no remainder is left.

## Illustration

Take for instance, multiplication of;

2 x 5 = 10

2 and 5 are considered to be factors of 10, and similarly, 10 is a multiple of 2 and 5.

## What is a factor?

As defined above, a factor is a number that divides a certain number to give out a whole number or integer.

For example, 2 is a factor of 8 because it divides 8 without leaving any remainder. 1 is the smallest factor of any number. The factors of 8 are therefore 1, 2, 4, and 8 themselves.

When listing factors of a given number, the first step is to identify all numbers that divide this particular number without a remainder. And to do so, 1 is listed as the first minimum factor. For example, the factors of a number are integers which divide that particular number without a remainder. The number 16 has five factors: 1, 2, 4, 8, and 16. If the number 16 is divided by any of the five numbers, then the resultant is a whole number

The factors of a number are the numbers that divide into it exactly.

For example, the number 12 has six factors:

16 / 2 = 16

16 / 2 = 8

16 / 4 = 4

16 / 8 = 2

16 / 16 = 1

### Square Numbers

Square numbers are numbers obtained by multiplying a number by itself. All square numbers have an odd number of factors.

For example, 4 has 3 factors, 16 has 5 factors

### Prime Numbers

A prime number is only divisible by 1 and itself. Therefore, prime numbers have only two factors. Prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 have only two factors.

## What is a Multiple?

A multiple is the product of two integers. 3 x 7 = 21, in this case, is a multiple of 3 and 7. All integers have the number itself and zero as their multiple.

Multiples of a given number are found by multiplying that number by number 1. The resultant answer after the multiplication is, therefore, the multiple of that number. ## Common Multiples

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

To find the common multiples, the following steps are followed.

• List all multiples for each number.
• Proceed with your list until at least two multiples are common to all the lists.
• Select the common multiples.

Multiples of 2 are obtained as:

2 x 1 = 2

2 x 2 = 4

2 x 3 = 6

2 x 4 = 8

2 x 5 = 10

2 x 6 = 12

2 x 7 = 14

And so forth. Multiples of 2 will be: 0, 2, 4, 6, 8 or 10.

Multiples of 3 are:

3 x 1 = 3

3 x 2 = 6

3 x 3 = 9

3 x 4 = 12

and so on.

Multiples of 5 include:

5 x 1 = 5

5 x 2 = 10

5 x 3 = 15

5 x 4 = 20

5 x 5 = 25

and so on. You can realize that every multiple of 5 ends with either 5 or 0.

## What are the differences between a factor and a multiple?

 Factor Multiple 1 A list of numbers, each of which can divide a certain number without leaving a remainder. A list of number which is aresult of a multiplication of the number 2 A number that can be multiplied with a particular integer to get another integer The product is obtained by multiplication of the number by an integer. 3 Factors are a number are limited Multiples are endless 4 Factors are generally less or equal to the number Multiples are greater or equal to the given number. 5 Factors are obtained  by division Multiples s are obtained by multiplication

Example 1

Which of the following statements are true or false?

1. 9 is a multiple of 3
2. 5 is a factor of 15.
3. 7 is a multiple of 21.
4. 13 is a factor of 25

Solution

9 is a multiple of 3 is true because 3 x 3 = 9.

5 is a factor of 15 is true because: 5 x 3 = 15.

7 is a multiple of 21 is false because many are always greater than equals that number.

13 is a factor of 25 is false because you can’t multiply a whole number by 13 to get 25.

Example 2

“10 is a factor of 50”? Which of the following phrases are true to this statement.

1. 50 is divisible by 10
2. 10 is divisible by 50
3. 10 is a multiple of 50
4. 50 is a multiple of 10

Solution

Factors are numbers you multiply to obtain larger ones, so it true to say 50 is divisible by 10. Multiples are bigger numbers obtained when you multiply two factors, so it possible to say that 50 is a multiple of 10.

Hence, only options a and d are correct.

Example 3

Identify the are true or false statement from the following list.

1. 5 is a factor of 105
2. 3 is a multiple of 121
3. 88 is a multiple of 9
4. 11 is a factor of 121

Solution

1. True, because 5 × 21 = 105.
2. False, because 121 is a multiple of 3, and so, 3 is a factor of 121.
3. This statement is not true because 88 is not divisible by 9.
4. The statement is true because: 11 x 11 = 121.

Example 4

List down all factors of 1000.

Solution

The factors of 1000 are: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 125, 200, 250, 500 and 1000.

Example 5

List down all multiples of 8 upto 1000?

Solution

The multiples of 8 upto 1000 include:

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96,104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192,200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296,304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392,400, 408, 416, 424, 432, 440, 448, 456, 464, 472, 480, 488, 496,504, 512, 520, 528, 536, 544, 552, 560, 568, 576, 584, 592,600, 608, 616, 624, 632, 640, 648, 656, 664, 672, 680, 688, 696,704, 712, 720, 728, 736, 744, 752, 760, 768, 776, 784, 792,800, 808, 816, 824, 832, 840, 848, 856, 864, 872, 880, 888, 896,904, 912, 920, 928, 936, 944, 952, 960, 968, 976, 984, 992, 1,000.

Example 6

The number 16 is divisible by 4, so which of these two numbers is a multiple and a factor and why?

Solution

16 is a multiple because it is bigger, whereas 4 is a factor because it is smaller.

### Practice Questions

1. Identify the statements that are true as “$15$ is a multiple of $3$.”

2. What are the three multiples of $90$ that lie between $400$ and $900$, and whose tens digit is odd?

3. True or False: There are ten multiples of 100 up to 1000.

4. Which two multiples of $60$, when rounded off to the nearest hundred, give $500$?

5. If Mike’s age is a multiple of $6$ in 2020 and a multiple of $5$ in 2021, what is the possible age(s) of Mike now?

6. True or False: There are twelve multiples of 99 up to 1000.

7. Which of the following is a two-digit even number which is a multiple of both $6$ and $14$?