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. 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)
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 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

Various Types of Factors
Even factors:
Even factors are the factors of number, which are divisible by 2.

Example 1: Find the number of even factors of 58800?
Solution: We first factorize 58800.
58800 = 2⁴ 3¹5²7²
In this case we have to find number of even factors, an even factor is divisible by 2 or we can say smallest power of 2 should be 1 not 0.
Hence a factor must have
2^{(1 or 2 or 3 or 4)}  — 4 factors

3^{(0 or 1 )}       — 1+1=2 factors

5^{(0 or 1 or 2)} — 1+2=3 factors

7^{(0 or 1 or 2)} — 1+2=3 factors

Hence total number of even factors = (4)(2)(3)(3) = 72
Hence, number of even factors of a number 
N=2paqbrcsare p(q + 1)(r + 1)(s + 1)N=2paqbrcsare p(q + 1)(r + 1)(s + 1)

Odd factors: 
Odd factors are those factors, which are not divisible by 2.

Example 2: Find the number of odd factors of 58800.
Solution: We first factorize 58800.
58800 = 2⁴ 3¹5²7²
Total number of factors of 58800 is 5x2x3x3=90 and in previous example we calculated total number of even factors is 72.
Number of odd factors = Total number of factors – Number of even factors
=90- 72 = 18

Alternate way: Since odd factors should have power of 2 as 0. Hence odd factor must have
2^{(0 )}  ———- 1 factor

3^{(0 or 1 )}     — 1+1=2 factors

5^{(0 or 1 or 2)} — 1+2=3 factors

7^{(0 or 1 or 2)} — 1+2=3 factors

Total number of odd factors = (1 )(2)(3)(3) = 18
Hence, number of odd factors of a number
N=2paqbrcsare p(q + 1)(r + 1)(s + 1)N=2paqbrcsare p(q + 1)(r + 1)(s + 1)

Perfect square factors:
If a number is perfect square then its prime factors must have even powers.

Example 3: Find the number of factors of 58800 which  are perfect square?
Solution:
We know that for a number to be a perfect square, its factor must have the even number of powers.
We first factorize 58800.
58800 = 2⁴ 3¹5²7²
Hence perfect square factors must have
2^{(0 or 2 or 4)}—– 3 factors

3^{( 0 )}     —–   1  factor

5^{(0 or 2)} ——- 2 factors

7^{(0 or 2)} — 2 factors

Hence, the number of factors which are perfect square are 3x1x2x2=12

Remember 
-If number of factors is odd then the number is a perfect square and vice versa  is also true i.e. if a number is a perfect square then number of factors is  odd.
This is because if number is a perfect square then p, q, and r are even and hence 
(p + 1) (q + 1) and (r+ 1) are odd and so product of these numbers is also an odd number. 
-If number of factors is even then number is not a perfect square.