This article will help you in understanding the methods and process of factoring the numbers. It also has tremendous applications in applied and theoretical physics.
What are Factors of Number?
In mathematics, a factor of a number is defined as the algebraic expression that divides any given number evenly, with its remainder being zero. In other words, it is also known as the product of multiple factors.
What are the Factors?
Factors are defined as the number which is multiplied by a number to get the original numbers.
Think we have a number 8 the factors are 2,4 because the product of 2 and 4 gives us 8. The other factors are 8 and 1.
The factors can also be -2, -4, and -1, -8 since the multiplication of two negative numbers is positive.
So, in total, there are factors 1,2,4,8, -1, -2, -4, -8.
Negative numbers are not considered as a factor mostly. Numbers that are in the fraction are not considered as a factor of any number.
2 will be a factor of every even number.
Numbers that have 5 in units place will have 5 as the factors.
Finding Factors of Numbers
There are three formulas that are considered as the best way to find factors of a number.
Numbers of factors
Product of factors
Sum of factors
Presume we have a natural number N for which we need to find the factors. If we convert the natural number N into the product of prime numbers (prime numbers are those numbers which are divisible by 1 and by itself and not by other numbers 0 and 1 are not considered as a prime number) then
Here X, Y, Z are the prime numbers, and l, m, n are their respective powers.
So, the total number of factors for N is,
(l+1) (m+1) (n+1)
The formula to find the sum of the factors is given by,
[(Xl+1 – 1)/X-1] × [(Ym+1 – 1)/Y-1] × [(Zn+1 – 1)/Z-1]
Product of factors is given by,
Number of factors /2
Example: Find the total number of factors, the sum of factors, a product of factors of 60.
Solution: First we will calculate all the factors of 60
Factors of 60 = 2×30
60= 22×31 ×51
Here X=2, Y=3, Z=5
l= 2, m=1, n=1
total number of factors of 60
= (l+1) (m+1) (n+1)
= (2+1) (1+1) (1+1)
Sum of factors of 60=
[(Xl+1-1)/X-1] × [(Ym+1-1)/Y-1] × [(Zn+1-1)/Z-1]
= [(23-1)/2-1] × [(32-1)/3-1] × [(52-1)/5-1]
Product of factors of 60= Number of factors /2
Steps to Find Factors of a Number
Choose a number.
Note down all the common factors of that number.
Prepare the factors of the number that we have got in step 2 until we get the prime numbers.
Note down all the factors that you have got.
List down all the unique factors that you have got.
To find all factors of a number, assume we have a number 18.
The factors which we will get in step 2 are (1×18), (2×9), (3×6).
Note that we will take only positive factors and not a negative one.
From step 3 (2×9) will be (2×3×3) and (3×6) will become (3×2×3)
So, the unique factors of a number 18 are 1,2,3,6,9,18.
Prime Factorization is a very simple process to get the factors. In this method, the number given is written as the product of the prime numbers.
How to Find Factors of a Large Number?
For the calculation of the factors of a large number, first, divide that number by 2 which is the least prime number.
In case the number is not divisible by 2, then go for the one which is not a prime number. Check if it is divisible or not. Repeat this process until you get the final value as 1.
Example: We have a number 1230.
Step 1: Firstly, we will divide it by 2, so we get 615. Now, this number can not be divided by 2.
Step 2: In this, case, we go for the next divisible number that is 3. After dividing the number 615 by 3, we get 25.
Step 3: Now, 25 cannot be divided by 3. Now for the next prime number that is 5. The result we get is 5 (25/5) which can again be divided by 5 giving us the expected result of 1 at last.
So, the factors of 1230 are 1, 2, 3, 5, 10, 25, 123, 246, 410, 615, 1230.
Greatest Common Factor
Greatest Common Factor is defined as the highest common factor between the two given numbers.
Think we have two numbers 10 and 20.
Factors of 10 are 1,2,5,10.
Factors of 20 are 1,2,4,5,10,20.
So, the common factors between them are 1,2,5,10. And the highest factor is 10. Therefore, as a result, the greatest common factor of 10 and 20 is 10.
Shortcut to Find Factors of a Number
The fastest way to find the factors of a number is to divide the number by the smallest prime number (bigger than 1). This needs to go into it evenly without any remainder. Go on with this process with each number you get, until you reach 1.