If $A,B$ are factors of $2^6 3^4 5^2,$ how many values of $|A-B|$ are possible?

Question

Let $x=2^6 3^4 5^2$, then how many distinct values of $|A-B|$ are possible where $A, B$ are the factors of $x$?

How to approach this problem?

Answer 1

There are $7\times5\times3=105$ divisors of $x$. There are at most $\binom{105}{2}+1=5461$ non-negative differences among the elements. $\binom{105}{2}$ computes the number of pairs of distinct factors and $1$ is for $0$, the difference between a factor and itself. There may be fewer than this many because some differences may be repeated.

To compute the actual number of differences, one could use a program like Mathematica. I used this code

d = Divisors[2^6 3^4 5^2]; Length[Union[Abs[Flatten[Outer[Subtract, d, d]]]]]

to compute that there are $2856$ distinct non-negative differences.