Basics of Factors:
What are factors? What does the term mean? If we go by the Latin translation of the word, it means ‘who/which acts’. Well, in our mathematical sense, factors are the ones that act for sure, but in this case they act on numbers.

Mathematically speaking: If any integer, say P, is divisible by another integer say Q an exact number of times then P is said to be a multiple of Q and Q is the factor of P.

Number of Factors of a given number:
Number of factors can be expressed by following steps:

  1. First write down the number in prime factorisation form i.e. ap bq cr (where a,b,c, are prime numbers and the p,q,r are natural numbers as their respective powers)
  2. Number of factors can be expressed as (p+1)(q+1)(r+1).
  3. Here factors include 1 and the number itself.

The points above explained through a example:
Let us take an example of a number N = 38491200=26 37 52
Now observe some facts about the number of factors (we will solve the problem step by step):
Step 1: Prime factorisation, so N=38491200=26 37 52 11
Power of 2 as 2, 21 ,22  ,23,24,25,26( 6+1=7)ways ,
Power of 3 as 3, 31 ,32 ,33,34,,35,36,37( 7+1=8)
Power of 5 as 50 , 51 ,52 ( 2+1=3)ways
Power of 11 as 110 , 111  ( 1+1=2)ways
Step 2:Hence, the number of factors is given by (6+1)(7+1)(2+1)(1+1)=7x8x3x2=336

Example 1:Find the number of factors of 24315272.
Solution: As we can see the above number has 2,3,5,7 which all are prime numbers and they have 4,1,2,2 as their powers so the number of factors of the given number are (4+1)(1+1)(2+1)(2+1)= 90
 
Example 2:Find the number of factors of 1440 ?
Solution: We first factorize 1440.
1440 = 253251
2,3,5 are prime numbers and they have 5,2,1 as their powers so the number of factors of the given number are (5+1)(2+1)(1+1)= 36




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