The factors of any number are the numbers that can divide a larger number with a whole number as the quotient and the remainder as zero. On multiplication of all the factors of a number, we get the number itself. The prime factors of a number are the smallest prime numbers that when multiplied with each other constitute the number itself.
For example, 32 when divided by 4, gives us the quotient 8, and a remainder that’s 0. This implies that 4 is a factor of 32. Also, we know that 32 is divisible by 2, 3, 4, 8, 16 and 32, which implies that all these numbers are factors of the number 32.
We can write any whole number as the product of two factors. These two factors are known as the factor pairs of the whole number. From the previous example, 4 × 8 = 32 implies that 4 and 8 form a factor pair of 32.
Suppose we want to make different rectangles with an area of 24 square units. Then what are the possible measures of length and width of these rectangles?
We will use the area model to find the solution. We draw rectangles of a given area of 24 square units with different sets of measurements for the length and width of these rectangles. In the area model, the product of the measure of length and width of each rectangle gives the area of 24 square units. Hence, we can say that each pair of length and width measurements forms a factor pair of 24.
Now, 1 × 24 = 24 makes a rectangle of area 24 square units, where 1 unit and 24 units are the measure of the length and width. Therefore, 1 and 24 is a factor pair of 24. Similarly, we know that 12 and 2, 8 and 3, 6 and 4 are also the factor pairs of 24.
Hence, 24 units and 1 unit, 12 units and 2 units, 8 units and 3 units, 6 units and 4 units are the possible sets of length and width measurements of rectangles with the given area of 24 square units.
Note: The multiplication equations, 4 × 6 = 24 and 6 × 4 = 24, both give 4 and 6, and 6 and 4 as two factor pairs. However, we consider it as a one factor pair i.e. 4 and 6. This is similar for other factor pairs also.
When two whole numbers are multiplied, their product is known as a multiple of the two whole numbers. For example, if we multiply 5 by 2, then the product is 10. So we say that 10 is a multiple of 5 and 2.
Any whole number is a multiple of each of its factors. The factors of 12 are 1, 2,3,4 and 6, so we can say that 12 is a multiple of 1,2,3,4 and 6. The fact table of a number can be used to find its multiples.
For example: Is 42 a multiple of 7?
One way to find the answer is to list the multiples of 7 and search for 42.
Multiples of 7 are:
Therefore, 42 is a multiple of 7.
Another way is to first find out whether 7 is a factor of 42 when we use division.
42 ÷ 7 = 6
This means 7 is a factor of 42. Therefore 42 is a multiple of 7.
Example 1: Is 22 a multiple of 3?
The multiples of 3 are, 3, 6, 9, 12, 15, 18, 24, 27……
In the list of the multiples of 3, we see that the multiple next to 18 is 21. This implies that 22 is skipped between them. Therefore, 22 is not a multiple of 3.
Example 2: Is 8 a factor of 56?
From the fact table of 8 we know that,
8 × 7 = 56
This implies that 56 is a multiple of 8. Therefore, 8 is a factor of 56.
Another way to solve this problem:
The multiples of 8 are, 8, 16, 24, 32, 40, 48, 56, 64….
Therefore number 56 is a multiple of 8. This implies 8 is a factor of 56.
Draw area model rectangles to find the factor pairs of 16.
We find the side lengths of as many different rectangles with an area of 16 square units as possible. The side lengths of each rectangle are one of the factor pairs of 16.
We see that three different rectangles with an area of 16 square units are possible with length and width as 16 units and 1 unit, 8 units and 2 units, 4 units and 4 units.
Therefore, the factor pairs of 16 are 1 and 16, 2 and 8 and, 4 and 4.
Example 4: A gardener buys 27 flowering plants. He wants to organize the plants into a rectangular array. How many different arrays can he make?
To find the number of rectangular arrays that the gardener can make, find the number of factor pairs of 27.
The factor pairs of 27 are:
The factor pairs of 27 are 1 and 27, 3 and 9. So there are two factor pairs of 27.
As we can use one factor pair to make two different rectangular arrays as shown in the image.
So, there are 2 × 2 = 4 ways to plant the flowering plants in a rectangular array.
Example 5: Name two numbers that are each a multiple of both 3 and 5. What do you notice about these two multiples?
We write the multiples of 3 as,
We write multiples of 5 as,
The common multiple of 3 and 5 are 15, 30, 45..etc.
If we observe these multiples, we can say that the common multiples of 3 and 5 are multiples of the product of three and five, such as 15, 30, and so on.
A common multiple is a number that is a multiple of more than one number. For example, the factors of 20 are 1, 2, 4, 5, 10. So, 20 is a common multiple of 1, 2, 4, 5 and 10.
1 is the only number whose only factor is 1 itself.
The proper factors of a number are the factors other than 1 and the number itself . For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. Therefore the proper factors of 12 are 2, 3, 4, and 6 (excluding 1 and the number itself, which is 12).