Number of integral solutions?

[SilverSprite]

What is the number of integral solutions to |x||y|z|=12?

[krtxmrtz]

The answer is 144.

If, first, you consider only the positive solutions, you have the following possibilities for the products:

Case (1): 12 x 1 x 1
Case (2): 6 x 2 x 1
Case (3): 4 x 3 x 1
Case (4): 3 x 2 x 2

and their permutations. The number of these is calculated as:

P = N! / (r1! x r2! x ...)

where N is the number of factors (3) and r1, r2... are the number of times they appear in the product, so:

Case (1): P = 3! / (1! x 2!) = 3
Case (2): P = 3! / (1! x 1! x 1!) = 6
Case (3): P = 3! / (1! x 1! x 1!) = 6
Case (4): P = 3! / (1! x 2!) = 3

which add up to 18.

Now, to include the negative solutions, each one of these18 possible permutations of the original products can be formed with all the possible permutations of positive and negative factors, i.e. the permutations (including repated values) of 2 elements (+ and -) taken in groups of 3 (such as ++-, -+-, etc). Their number is calculated as:

P2 = 2**3 = 8

So, finally, the overall number of solutions will be P1*P2 = 144

[SilverSprite]

Wow i didnt think that answer needed that much work. Anyways that is correct.