Factors  Atreya Roy

Author: Atreya Roy is pursuing his BTech From Kalyani Government Engineering College, Bengal.
Factor of a number is a number smaller or equal to it which divides the number wholly. Hence factor of a number cannot exceed the number itself. Factors are always expressed as the product of primes present in that number.
Example : Find the factors of 30
Solution : 30 = 2 * 3 * 5
In simple terms we can say that the factors of 30 = 1,2,3,5,6,10,15,30 : Total 8 factors.
Let us look into a formula we can keep in mind so that we can calculate the number of factors fast.
If the Number is represented in the form N = (a ^{p})*(b^{ q})*(c^{ r})*(d^{ s })…. Where a,b,c,d.. are the primes present in the number and p,q,r,s… are their respective exponents.
So the number of factors in the number = (p+1)*(q+1)*(r+1)*(s+1)……. And so on.
Example : Find the factors of 30
Solution : 30 = 2^1 * 3^1 *5^1
P=1
Q=1
R=1
Hence the number of factors of 30 = (1+1)(1+1)(1+1) = 2*2*2 = 8 factors.
Factors are of two types :
 Even factors : the factors which are divisible by 2
 Odd factors : the factors which are not divisible by 2
Example :
Find the number of even factors in the number 3600
Solution :
Break 3600 into its prime factors. 3600 = 100*36 = 2^4*3^2*5^2
Hence the total number of factors of 3600 = (4+1)*(2+1)*(2*1) = 45
Out of these how many are even. For being even factors the factors must have atleast one 2 in them. Hence out of the four 2s present. Take out one. We are left with :
2^3*3^2*5^2
Hence, with these factors if we multiply the 2 we took out, we will get the total even factors.
Total even factors = (3+1)*(2+1)*(2+1) = 4*3*3 = 36
From this we can also find out the odd factors.
Total Factors = Even Factors + Odd factors.
Hence if the total number of factors = 45 and even factors are 36 then 4536 = 9 odd factors are present in the number.
Note : To find the number of odd factors present in a number we can also calculate them by removing all the 2s present in the number.
Problem :
We are given a number 64800. What are the total number of :
 Prime Factors
 Composite Factors
 Odd factors
 Even Factors
Answer :
Breaking 64800 into its factors, we get : 2^5 * 3^4 * 5^2 .
 Prime factors : 2,3,5 : hence 3 factors
 Total factors = (5+1)*(4+1)*(2+1) = 6*5*3 = 90
Hence composite factors = total factors – prime factors1 (since 1 is neither prime nor composite) = 9031  Odd Factors = factors in the number : 3^4 * 5^2 = (4+1)*(2+1 ) = 15
 Even Factors = total factors – odd factors = 9015 = 75