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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. In case of any queries you can reach him at [email protected]

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

Answer :

Breaking 64800 into its factors, we get : 2^5 * 3^4 * 5^2 .

  1. Prime factors : 2,3,5 : hence 3 factors
  2. Total factors = (5+1)*(4+1)*(2+1) = 6*5*3 = 90
    Hence composite factors = total factors – prime factors-1 (since 1 is neither prime nor composite) = 90-3-1
  3. Odd Factors = factors in the number : 3^4 * 5^2 = (4+1)*(2+1 ) = 15
  4. Even Factors = total factors – odd factors = 90-15 = 75