Factor of a Number
The factors of a number are numbers that divide it completely. Alternatively, we can view factors of a specific number as numbers that can be multiplied by another number to get the given number as the product. We will get zero as a remainder when we divide a number by any of its factors. On the other hand, we will obtain a value as a remainder when we divide a number by a number that is not its factor. The factors of a number can be positive or negative integers.
For example: The positive factors of 64 are 1, 2, 4, 8, 16, 32, and 64.
The negative factors of 64 are – 1, – 2, – 4, – 8, – 16, – 32 and – 64.
Factor Pairs
A factor pair of a number is the set of two of its factors, such that their product is the number itself. The factor pair can be negative or positive.
For example, the factor pairs of 48 are given below.
Positive Factor Pairs of 48 | Negative Factor Pairs of 48 |
|---|---|
(1, 48) | (- 1, - 48) |
(2, 24) | (- 2, - 24) |
(3, 16) | (- 3, - 16) |
(4, 12) | (- 4, - 12) |
(6, 8) | (- 6, - 8) |
(8, 6) | (- 8, - 6) |
(12, 4) | (- 12, - 4) |
(3, 16) | (- 3, - 16) |
(2, 24) | (- 2, - 24) |
(48, 1) | (- 48, - 1) |
How do we Obtain Factor Pairs?
Factor pairs of a number can be obtained by multiplying two numbers that result in the given number. We divide the given number from 1 onwards till the quotients become divisors, and at this point the factor pairs of a number have started to repeat.
Solved Examples
Example 1: What are the factor pairs of 18?
Solution:
The factor pairs of 18 are,
(1, 18), (2, 9), (3, 6), (9, 2), (18, 1), (- 1, – 18),
(- 2, – 9),(- 3, – 6),(- 6, – 3),(- 9, – 2) and (- 18, – 1)
Example 2: Annie baked 56 cookies with her mom. She wants to decide the best way to put the cakes in rows and columns, to make them look attractive. Help Annie find the best way to arrange the cookies.
Solution:
Factor pairs of 56 would give us ways to divide the cookies in the form of rows and columns.
Factor pairs of 56 are:
1 \( \times \) 56 = 56
2 \( \times \) 28 = 56
4 \( \times \) 14 = 56
7 \( \times \) 8 = 56
(1, 56) , (2, 28), (4, 14), (7, 8) are the factor pairs of 56.
Hence, 56 cookies would look best when arranged in 7 rows and 8 columns or 8 columns and 7 rows.
Example 3: Tom was studying geometrical shapes and found a rectangle of area 36 square centimeters. What are the possible dimensions of the length and width of the rectangle? Answer in integers.
Solution :
The area of rectangle = \(l \times~w \) [Use formula]
36 = \(l \times~w \) [Substitute value]
The product of the length and width is 36.
The possible dimensions of length and width are the factor pairs of 36.
Dimension of rectangle | Factor Pairs of 36 |
|---|---|
| \(1 \times~36 \) | (1, 36) |
| \(2 \times~18 \) | (2, 18) |
| \(3 \times~12 \) | (3, 12) |
| \(4 \times~9 \) | (4, 9) |
| \(6 \times~6 \) | (6, 6) |
| \(9 \times~4 \) | (9, 4) |
| \(12 \times~3 \) | (12, 3) |
| \(18 \times~2 \) | (18, 2) |
| \(36 \times~1 \) | (36, 1) |
Composite numbers are those numbers that have a factor other than 1 and themselves, that is, a composite number has more than two factors.
A factor pair of a number is the set of two of its factors, such that their product is the number itself.
The factors of prime numbers are 1 and the number itself. So, there will be only one factor pair of prime numbers.
The division of any positive number by 1 results in zero as the remainder, and the quotient is the number itself. Hence, 1 is a factor of all positive numbers.