Basics of Factors

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. a^p b^q c^r(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 ,3²,3³,3^4,,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