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Factor pairs: Definition, Facts with Examples

factor pairs

What are factor pairs? The two positive/negative integers that give a number on multiplication are the factor pairs of that number. Suppose you multiply two numbers to obtain a product, then the numbers multiplied are the factors of the product. Also, a number is a factor in itself. We use the multiplication or division method to find factor pairs of a number.

Properties of Factors

Following are some important properties of factors of any number:

  • Every number has 1 as its factor, and every natural number is a factor of itself. So, one into the number is a factor pair of every number. 
  • The factor of a given number is less than or equal to the number. 
  • There are a finite number of factors of a given number.
Fascinating Facts: A square number has a factor pair wherein one factor is multiplied by itself. This factor is the square root of the given number. For example, 16 has the following factor pairs:1 × 162 × 84 × 4Here we can see that the third pair has the same number, 4 multiplied by itself. Thus, you can also recognize the square root of a number from the list of factor pairs, like here, the square root of 16 is 4. 

How to Find Factor Pairs?

Factor pairs can be calculated using both multiplication and division methods. 

Finding Factors using Multiplication

As the multiplication of two numbers gives a product, the two numbers become the factors of the product. Suppose we have to find the factors of a number, say N. We will write the multiplication equations that have two factors such that the resulting product is equal to N. Suppose we have to find the factor pairs of 12.

So, we will write the possible combinations as follow:

1 × 12 =12

2 × 6 = 12

3 ×  4 = 12

Thus, the factor pairs of 12 are:

  • (1 × 12)
  • (-1 × -12)
  • (2 × 6)
  • (-2 × -6)
  • (3 ×  4)
  • (-3 × – 4)

Finding Factors Using Division

To find factor pairs for a given number N using the division method, follow the below-given steps: 

  • Figure out numbers that are less than N.
  • Now, divide N from those numbers in a way that a resulting quotient is a whole number. 
  • Every divisor and quotient pair will form a factor pair for the number N. 

For example, we have to find the factor pairs of 8. The numbers less than 8 are 1, 2, 3, 4, 5, 6, 7, and 8.

Let us divide 8 by each of these numbers. 

8/1 = 8

8/2 = 4

8/3 = 2 (remainder 2)

8/4 = 2 

8/5 = 1 (remainder =3)

8/6 = 1(remainder = 2)

8/7 =  1(remainder =1)

Thus we can see that the factors of 8 are:

  • 1 ×  8
  • -1 × -8
  • 2 × 4 ( which is the same as 4 and 2)
  • -2 × -4
Fascinating Facts: Every natural number is the product of at least one-factor pair. 1 is a factor of every number. So, it is present in a factor pair of every number. For example: 1 x 6 = 6, 1 x 11 = 11, and so on. 2 is a factor of every even number. So, it is present in a factor pair of every even number. For example: 2 x 4 = 8, 5 x 2 = 10, 6 x 2 = 12, and so on.The divisor and quotient in the division both form a factor pair of the dividend if the remainder is zero.

Can pair factors include negative numbers?

It is interesting to note that although factors of a number cannot be negative numbers, their factor pairs can include negative numbers. Why does this happen?

A factor pair is a multiplication of two numbers to give a product. So, when two negative numbers multiply, their product is a positive number (Remember the rule: minus x minus = plus).

So, both the numbers in the factor pair should be either negative or positive to give a positive number as a product. 

For example: (-3 and -6) and (-2 and -9) form factor pairs for 18. 

Factor Pair of Prime Numbers

A prime number has only two factors, 1 and the number itself. So, any prime number will have a single factor pair consisting of positive numbers. It can also have a factor pair with negative numbers. 

For example, the prime number 17 has (17 × 1) and (-17 × -1) as its factor pairs.

Here is a list of prime factors from 1 to 100. Each has two-factor pairs that include 1 and the number itself when both are positive and also when both are negative. 

235711
1317192329
3137414347
5359616667
7173798397

What are factor pairs of 24?

We can find the factor pairs of 24 as given below:

1 × 24 = 24

2 × 12 = 24

3 × 8 = 24

4 × 6 = 24

Thus, the factor pairs of 24 are: 

  • (1 × 24)
  • (2 × 12)
  • (3 × 8)
  • (4 × 6 ) 
  • (-1 × -24)
  • (-2 × -12)
  • (-3 × -8)
  • (-4 × -6 ) 

What are factor pairs of 36?

We can find the factor pairs of 36 by multiplication method as given below:

1 × 36 = 36

2 × 18 = 36

3 × 12 = 36

4 × 9 = 36

6 × 6 = 36

When we start getting repeated numbers in multiplication, we should stop writing 36 as the product of other numbers.

Thus, the factor pairs of 36 are: 

  • (1 × 36)
  • (2 × 18)
  • (3 × 12)
  • (4 × 9 ) 
  • (6 × 6 )
  • (-1 × -36)
  • (-2 × -18)
  • (-3 × -12)
  • (-4 × -9 ) 
  • (-6 × -6 )

What are factor pairs of 48?

To find the factor pairs of 48 by multiplication method, follow the steps given below:

1 × 48 = 48

2 × 24 = 48

3 × 16 = 48

4 × 12 = 48

6 × 8 = 48

Thus, the factor pairs of 48 are: 

  • (1 × 48)
  • (2 × 24)
  • (3 × 16)
  • (4 × 12 ) 
  • (6 × 8 )
  • (-1 × -48)
  • (-2 × -24)
  • (-3 × -16)
  • (-4 × -12 ) 
  • (-6 × -8 )

What factors of 72 in pairs?

We will use the multiplication method as follows to the factor pairs of 72: 

1 × 72 = 72

2 × 36 = 72

3 × 24 = 72

4 × 18 = 72

6 × 12 = 72

8 × 9 = 72

Thus, the factor pairs of 72 are: 

  • (1 × 72)
  • (2 × 36)
  • (3 × 24)
  • (4 × 18) 
  • (6 × 12)
  • (8 × 9)
  • (-1 × -72)
  • (-2 × -36)
  • (-3 × -24)
  • (-4 × -18) 
  • (-6 × -12)
  • (-8 × -9)

Factor Pair Chart

The following chart has factor pairs for numbers up to 45, excluding prime numbers. 

Note: You must also include negative numbers of each pair when you write all possible factor pairs of any number.

NumberFactor PairsNumberFactor PairsNumberFactor Pairs
41 × 4
2 × 2
201 × 20
2 × 10
4 × 5
331 × 333 × 11
61 × 6
 2 × 3
211 × 21
3 × 7
341 × 342 × 17
81 × 8
2 × 4
221 × 22
2 × 11
351 × 355 × 7
91 × 9
3 × 3
241 × 24
2 × 12
3 × 8
4 × 6
361 × 362 × 183 × 124 × 96 × 6
101 × 10
2 × 5
251 × 25
5 × 5
381 × 38
2 × 19
121 × 12
2 × 6
3 × 4
261 × 26
2 × 13
391 × 39
3 × 13
141 × 14
2 × 7
271 × 27
3 × 9
401 × 40
2 × 20
4 × 10
5 × 8
151 × 15
3 × 5
281 × 28
2 × 14
4 × 7
421 & 42
2 & 21
6 & 7
161 × 16
2 × 8
4 × 4
301 × 30
2 × 15
3 × 10
5 × 6
441 × 44
2 × 22
4 × 11
181 × 18
2 × 9
3 × 6
321 × 32
2 × 16
4 × 8
451 × 45
5 × 9

Let us Summarize

You must keep in mind the following points related to factor pairs:

  • Factor pairs always consist of integers.
  • The factors of a number cannot be negative, but a factor pair can consist of two negative numbers. 
  • Prime numbers have only two-factor pairs. The number and 1 and the negative numbers of the same pair.

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