How many ways can you express 360 as product of 2 co-primes

Unordered : 2^(n-1)

Ordered : 2^n

How many factor pairs of 360 exist which are co-primes to each other

Let me explain the concept of factor pair with a small example.

let say you need co-prime factor pairs of 10. then what would be the answer?

(1,1), (1,2) ,(1,5), (1,10) , (2,5) ..

now lets decipher n generalize it

Factor Pairs of a number x = 2^m * 3^n * .....

Case - 1 (1,1 ) --> 1

Case -2 = ; (1,2^m) ; (1,3^n) ; (As HCF should be 1)

total arrangements = 2(m+n)

case - 3 = (1,2^m*3^n) (As HCF should be 1)

total arrangements = 2 * m * n

Case - 4 (2^m , 3^n) (HCF = 1)

total arrangements = 2 * m * n

So total ordered cases for factor pairs = 1 + 2(m + n) + 4mn = (2m + 1) * (2n + 1)

this is the general formula for factor pairs (ordered )

in case of x = 360

360 = 2^3 * 3^2 * 5

Ordered factor pairs = (2 * 3 + 1) * (2 * 2 + 1) * (2 * 1+ 1 ) = 105