Factors and Multiples are co-related terms. Yet they are quite different from each other.
So let’s start our discussion by going through each of the topics individually.
Let there be a number a and another number b
Suppose, if a X b =c
Then a and b are the factors of c.
Thus any whole number(a) which when multiplied to another whole number(b) gives another whole number(c). Then the first two number is said to be the factor of the last whole number obtained.
So, factors of a particular number, are those numbers that divide the number exactly without leaving a remainder.
Now, the term “whole number” must be noted here. Factors are only the numbers in the number system which have no value after the decimal point. This is done to maintain a finite number of factors for a particular factor.
Other than the whole number, the negative integers of the number system can also be a factor of a particular number, known as the negative factors.
Similarly, while calculating factors of negative numbers we get factors that can be positive or negative.
Note: Every number has at least two factors: 1 and the number itself.
1. Taking a positive integer at first,
What are the factors 12?
The factor pairs of 12 are:
1 X 12 = 12
2 X 6 = 12
3 X 4 =12
Hence the there are total 6 factors of 12:
1, 2, 3, 4, 6, 12
2. Now, Taking a negatve intiger, -8
The factor pairs of -8 are:
1 × -8 = -8
2 × -4 = -8
4 × -2 = -8
8 × -1 = -8
There are total 8 factors of 8, and they are : 1, 2, 4, 8, -1, -2, -4, -8
Let, us take another example to understand this topic more clearly.
Let’s say we have a number a and number b,
so if b = a X n, where n = any whole number
Then b is known to be the multiple of a.
Coming to the definition now,
Any number when multiplied by another whole number, gives a product. That product is defined as the multiple of each of those numbers.
They are nothing, but product results. They form separate sets of numbers for any individual base number.
Technically negative multiples exist in mathematics too.
Note: Any individual integers can have an infinite number of multiples
Now take few examples,
1. List 5 multiples of 3
The list is as follows:
3 X 1 = 3
3 X 2= 6
3 X 3 = 9
3 X 4 =12
3 X 5 =15
So, any five multiples of three are listed as: 3, 6, 9,12,15
2. List 4 multiples of -4
-4 x 1 = -4
-4 x -1 = 4
-4 x 2 = -8
-4 X -2 =8
Four multiples of -4 are -4, 4, -8, 8
(we can generate negative multiples easily by multiplying the base number with any negative integers)
Thus, from all of this, it is quite clear that ‘factors’ and ‘multiples’ are two different types of numbers. But they are correlated to each other.
If we take a x b = c
Then we can say that a and b are factors of c
And c is a factor of and c is a factor of b.