Factors

12 can be expressed in terms of its prime factors as 2² * 3¹.

-> Number of factors = (2+1) * (1+1) = 6

Note: The 6 factors are: 1,2,3,4,6,12

15876 = 2² * 3⁴ * 7²

-> Number of factors (divisors) = (2+1)*(4+1)*(2+1) = 45

However, these 45 factors also include 15876 as a factor. Hence there will be 45-1=44 factors excluding the number itself.

24 = 2³ * 3¹

-> 24 has two prime factors, 2 and 3

-> 24 can be written as a product of two co-prime factors in 2^(2-1) or 2 ways.

Note: The two ways of writing 24 as a product of two co-prime factors are (1*24) and (3*8)

Let N be a perfect square, such that N = √N * √N = a^p * b^q * c^r

Then N can be expressed as a product of 2 factors in ((p+1) *(q+1) *(r+1) +1))/2 ways

Now, 9216 = 96*96 = 3² * 2¹⁰

-> It can be expressed as a product of 2 factors in ((2+1)*(10+1)+1))/2 = 17 ways

8064 = 2⁷ * 3² * 7¹

-> Sum = ((2⁸-1)/(2-1)) * ((3³-1)/(3-1)) * ((7²-1)/(7-1)) = (255*26*48)/(2*6) = 26520

36 can be expressed in terms of its prime factors as 2² * 3²

-> Sum of all factors of 36 = (1+2+2²) (1+3+3²) = 91

Note:

The various factors of 36 are: 1,2,3,4,6,9,12,18,36

540 = 2² * 3³ *5

-> Total number of factors of 540 = (2+1) (3+1) (1+1) = 24

-> Number of odd factors will include all factors without 2 = Number of ways in which we can select powers of 3 and 5.

Now, powers of 3 can be selected in 4 ways i.e. 3⁰, 3¹, 3² and 3³ and powers of 5 can be selected in 2 ways i.e. 5⁰ and 5¹. Hence number of odd factors = 4 * 2 = 8. Also, number of even factors = 24-8 = 16