# Factors - Atreya Roy

Atreya Roy is pursuing his BTech From Kalyani Government Engineering College, Bengal and runs his own venture Esser Solutions. He love math and is a proponent of free education. Atreya love sharing new concepts and shortcuts with MBA aspirants and regularly organize online events to support them.

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 :

1. Even factors : the factors which are divisible by 2
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 45-36 = 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 :

1. Prime Factors
2. Composite Factors
3. Odd factors
4. Even Factors