Suppose we have a positive integer $r$. The square of $r$ must be expressed as the product of two other positive integers such that $$r^2=pq$$ $p,q,r\in \mathbb [0,a-1]$, where $a$ is an integer.
Now, for all such $r$, in how many ways can this be done? Can a general rule be found out such that we can find the aforementioned number of ways for any given $r$?
I was trying to solve another problem which required the solution of the above. I tried to find partitions in which some kind of general pattern might be present but I could not find any such thing.
The total number of ways is given in the book as $(a-1)^2$, without any explanation. I suspect that it is wrong.
You are counting the number of (positive integer) divisors of $r^2$, denoted $\tau(r^2)$.
Given a prime factorization of $r=p_1^{a_1}\cdots p_k^{a_k}$, then $$ \tau(r)=(a_1+1)\cdots(a_k+1) $$ and $$ \tau(r^2) = (2a_1+1)\cdots(2a_k+1). $$ Your $a$ does not come into this.