Factors are great tools to study more about a number. Note that factors of a number are the same thing as the divisors of that given number. For example, both 2 and 3 divide the number 6, so they are the factors of 6.
Thus it is enough to study the divisors of a number to find the factors of that number. The divisibility rules will help us in this context. Here we will learn about factors, how to find them, properties and examples.
Table of Contents
Definition of factors of a number
What are factors? A number x is called a factor of a number if x divides the given number completely without leaving any remainder. The collection of all such x’s is called the factors of the given number.
Note that the set of divisors of a number is the same as the factors of that number.
For example, 1, 2, 5, and 10 divide the number 10. No other number can divide 10. So by definition 1, 2, 5, and 10 are the factors of 10.
Remark: The factors of a number can be negative. See that -2 is also a factor of 4 as it divides 4. But, remember that only positive factors are considered while solving a problem if there is no mention of negative factors.
Factors of a Prime Number
At first, we recall what are prime numbers. A number is said to be a prime number if it is only divisible by 1 and that number itself.
From the definition of a prime number, it is clear that the only divisors of a prime number are 1 and the number itself. So the factors of a prime number are 1 and that number itself.
For example, we know that 7 is a prime number. So its factors are 1 and 7.
How to Find Factors of a Number?
Let x be a given number. To find the factors of x, we need to follow the below steps:
Step 1: At first, we will find all the possible ways to write x multiplicative. In other words, we express x as a×b in all possible ways.
Step 2: We will list all the different numbers that appeared in a×b in step 1.
Step 3: The numbers obtained in step 3 will be factors of the given number x.
Example: By the above method, we will now find the factors of 48. At first, we will write 48=a×b in all possible ways. Note that we have
48 = 1×48
48 = 2×24
48 = 3×16
48 = 4×12
48 = 6×8
Observe that there are no other ways to express 48 other than the above ones. We see that the different numbers involved in 48=a×b are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. So the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. So there is a total 10 number of factors of 48.
Properties of Factors
(i) Factors can be negative, but we usually mean positive factors without mention.
(ii) The number 1 has only one factor which is 1. The numbers other than 0 and 1 have at least two factors, namely 1 and that number itself.
(iii) Each whole number has a finite number of factors. Note that the number 0 has infinitely many factors as it is divisible by any natural number.
Formulas of Factors
Let N be a whole number. Suppose the below is the prime factorization of N:
N = px × qy × rz
where p, q, and r are different prime numbers and x, y, z are natural numbers.
Number of factors formula:
The total number of factors of N is equal to
(x+1)(y+1)(z+1).
Sum of factors formula:
The sum of factors of N is equal to
$\frac{p^{x+1}-1}{p-1} \times \frac{q^{y+1}-1}{q-1}\times$ $\frac{r^{z+1}-1}{r-1}$
Product of factors formula:
The product of factors of N is equal to
NTotal No. of factors/2
