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:
- 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)
- Number of factors can be expressed as (p+1)(q+1)(r+1).
- 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 20 , 21, 22 ,23,24,25,26( 6+1=7)ways,
Power of 3 as 30 , 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