Hello pinaki

You are not clear with what the question is asking. I am trying here , let me begin

He is asking total number of factors possible. let suppose we have number 20.

Now tell me possible factors of 20 , good if you're saying 1,2,4,5,10,20 (brute force approach).

if you reform 20 in form of prime number then it would be like 5*2^{2 }

Now out of these possible factors i will tell you how 1,2,4,5,10,20 are possible

1-->>>possible when neither you take 5 nor 2

2 -->>> possible when you didn't take 5 but take one 2

4 -->>> possible when you didn't take 5 but take two 2's

5 -->>> possible when you take one 5 but no 2

10 -->>> possible when you take one 5 and one 2

20 -->>> possible when you take one 5 and two 2's.

so using brute force you solve it but what if number is large enough like 524678 then you have to derive some method so here from prime factor form i will tell you how you approach this.

let we have factored in prime form and that is 3^{4}5^{2}11^{7}13^{8 }

Now we have to choose possible 3,5,11,13

It's like choose any number of fruit out of 4 apples (all identical)and 3 mangoes (all identical) so either you will take no apple and no mango so that is one way , or no apple but one mango that is another way and so on so ultimately

ways to choose fruit = ways to choose apple*ways to choose mango

take 0 apple or 1 or 2 or 3 or 4 so 5 ways

take 0 mango 0r 1 or 2 or 3 so 4 ways ultimately (4+1)(3+1) ways

we have to solve given problem in same way

NOw about your doubt 3^{0 }5^{0 }11^{0 }13^{0 }means we didn't choose 3 , didn't 5,11,13 so then 1 is such a number when none choose. it doesn't mean we have chosen one by 3^{0 }only no to make 1 we need all number's power 0.

tell me if you didn't get yet.