 # Factors of a Number

Factors of a number are defined as numbers or algebraic expressions that divide a given number/expression evenly.  We can also say, factors are the numbers which are multiplied to get another number. For example, 1, 3 and 9 are the factors of 9, because 1 × 9 = 9 and 3 × 3 = 9. Here, the concepts of factors are explained which will help to understand how to find the factors and know the prime factors of some common digits. Here we will discuss finding factors, formulas to find the number of factors, product and sum of factors.

## Definition of Factors of a Number

The factors of a number are defined as the number which can be multiplied to get the original number. By multiplying two factors of a number, a product is obtained which is equal to the original number. It should be noted that factors of any number can be either positive or negative. For example, in the case of 6, the factors can be 2 and 3 as 2 × 3 gives 6. Here, 2 and 3 are factors while 6 is the product. The other factors of 6 are 1 and 6, etc. We can also consider -1, -2, -3 and -6 as the factors of 6, because when we multiply any two negative numbers, it results in positive number, such as;

-1 × -6 = 6

-2 × -3 = 6

Hence, the factors of 6 in total are 1,2,3,6,-1,-2,-3 and -6.

But normally, the factors are considered to be only positive numbers.

Point to remember: Fractions could not be considered as factors for any number.

### Factors Formulas

There are basically three types of formulas considered for factors. They are:

• Number of Factors
• Product of Factors
• Sum of Factors

Let us assume N is a natural number, for which we need to find the factors. If we convert N into the product of prime numbers by prime factorisation method, we can represent it as;

N = Xa × Yb × Zc

where X, Y and Z are the prime numbers and a, b and c are their respective powers.

Now, the formula for the total number of factors for a given number is given by;

• Total Number of Factors for N = (a+1) (b+1) (c+1)

The formula for the sum of all factors is given by;

• Sum of factors of N = [(Xa+1-1)/X-1] × [(Yb+1-1)/Y-1] × [(Zc+1-1)/Z-1]

The formula for the product of all factors is given by;

• Product of factors of N = NTotal No. of Factors/2

Example: Find the total number of factors of 90 along with sum and product of all factors.

Solution: Write the prime factorisation of 90 first.

90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5

90 = 21 × 32 × 51

Here, X = 2, Y = 3, Z =5 and a = 1, b = 2, c = 1

Therefore, total number of factors of 90 = (a +1)(b+1)(c+1) = (1+1)(2+1)(1+1) = 2 × 3 × 2 = 12

Sum of factors of 90 = [(21+1-1)/2-1] × [(32+1-1)/3-1] × [(51+1-1)/5-1] = 3/1 × 26/2 × 24/4 = 3 × 13 × 6 = 234

Product of factors of 90 = 90Total factors of 90/2 = 9012/2 = 906

### How to Find Factors of a Number?

Knowing how to calculate factors of a number is extremely crucial in maths. The steps to find the factors of a number are given below in a very easy to understand way. An example is taken to make the explanation easier.

• Step 1: Choose a number (say, 16)
• Step 2: Write the common factors of 16 which will include (16 × 1), (-16 × -1), (8 × 2), (-8 × -2), (4 × 4), and (-4 × -4).
• Step 3: Further factor the factors until a prime number is reached. In this case, 8 can be factored further.
• Step 4: Write down all the factors again. The (8 × 2) will now become (4 × 2 × 2).
• Step 5: Write all the unique number that is obtained.

So, the factors of 16 will be 1, 2, 4, 8, 16, – 1, – 2, – 4, – 8, and – 16. Here, the positive factors of 16 are only 1, 2, 4, 8, and 16.

Another Example:

Consider the number as 80.

80 = 10 × 4

= (5 × 2) × 8

=(5 × 2) × (4 × 2)

= (5 × 2) × (2 × 2 × 2)

Now, the factors of 80 will all the combination from 5 × 2 × 2 × 2 × 2 and 1 itself (as 1 × 80 = 80). So, the positive factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40, and 80. It should be noted that there will also be negative factors whose count have to be even.

### How to Calculate Factors of Large Numbers?

To calculate the factors of large numbers, divide the numbers with the least prime number i.e. 2. If the number is not divisible by 2, move to the next prime numbers i.e. 3 and so on until 1 is reached. Below is an example to find the factors of a large number.

Example: 1420

 Steps Prime Factors Product Step 1: Divide by 2 2 710 Step 2: Again Divide by 2 2 355 Step 3: Divide by 5 71

In step 3, a prime number is obtained as a product and so, the process is stopped. The factors will be all the multiples of 1, 2, 2, 5, 71, 355, 710. Now, the positive factors of 1420 will be 1, 2, 4, 5, 10, 20, 71, 142, 284, 355, 710, and 1420.

In the same case, if only prime factors are considered, it is called the prime factorization of that number. In this way, it is easy to factor a number and know its factors and prime factors.

### Factors of Some Common Numbers List

Below is a list of common numbers with their factors and prime factors. Each of these links will include the process of factoring of any number along with all the factors including the prime factors of that number.

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