Question

In how many ways can a number 6084 be written as a product of two different factors ? op 1: 27

Answer 1

First find the prime factors of 6084 
2*3042 
2*2*1521 
2*2*3*507 
2*2*3*3*169 
2*2*3*3*13*13 

2^2*3^2*13^2 

Number of factors = (2+1)*(2+1)*(2+1) = 3*3*3 = 27 
Using each of these factors, we can write 6084 as a product of 2 factors. 
Half of these will remain same. 

Let us take a simple example, 6 
number of factors = 4 
1,2,3,6 
can be written as 
1*6 
2*3 
We cant repeat 6*1 and 3*2 as they are already taken. 
Hence only 2 ways. 

Another example: take 9. 
Number of factors = 3 
1,3,9 
can be written as 
1*9 only. 
Hence only 1 way. 

So it turnsout ot be number of factors/2 if number of factors is even and 
(number of factors-1)/2 if number of factors is odd. 

Therefore: 27-1/2 = 26/2 = 13 .