Is there a general method to count number of ways a number N can be written as a product of k Integers (positive or negative).
The specific question I encountered was to count in how many ways I can write 144 as a product of 3 positive integers and so I wonder if there is any general method. Like I know that $144 = 2^{4}3^{2}$ and so there would be $(4+1)(2+1) = 15$ factors in total but then how to count number of ways I can divide $144$ into product of three factors.
Clarification: Order of factors don't matter i.e. $1⋅12⋅1$ is same as $12⋅1⋅12$ and so on for other combinations. Also we consider 1 as a factor i.e. $1⋅12⋅12=144$ is to be considered.
If the prime factortization of $N$ is $$N=\prod_i p_i^{a_i}$$ with $a_i\in\Bbb N_0$ (and distict primes $p_i$), then to write $N$ as product of $k$ factors, we need do write each $a_i$ as sum of $k$ non-negative integers (which are then the exponents of $p_i$ in the corresponding factor). Using "stars and bars", we conclude that there are $$ \prod_i {a_i+k-1\choose k-1}$$ ways to write $N$ as product of $k$ positive integers.
If we allow negative factors, we can pick $k-1$ signs and then the last sign is determined, so now we count $$ 2^{k-1}\prod_i {a_i+k-1\choose k-1}$$ ways.