Determine the number of integral solutions of |x| * |y| * |z|=12.?

Question

Thanks!

Answer 1

Well, the easy part is finding the four ways to get the integers to multiply to 12:

1, 1, 12
1, 2, 6
1, 3, 4
2, 2, 3

The tricky part is finding the exact number, since you have permutations to consider, along with the fact that any number can be positive or negative.

First, for the permutations: for the two with three different integers, there are six ways to arrange the numbers. For example, the six arrangements for 1, 3, 4 are: (a) 1, 3, 4 (b) 1, 4, 3 (c) 3, 4, 1 (d) 3, 1, 4 (e) 4, 3, 1 (f) 4, 1, 3. Allowing for each number to be positive or negative gives 2 * 2 * 2 = 8 solutions for each of the six ways. So, there are 8 * 6 = 48 answers.

Second, for the two with two integers the same (i.e. 1, 1, 12 and 2, 2, 3), there are three ways to arrange the numbers. For example, the three arrangements for 2, 2, 3 are (a) 2, 2, 3 (b) 2, 3, 2 (c) 3, 2, 2. Allowing the same eight ways to set the signs, that gives 24 answers.

So, in total, there are 48 + 48 + 24 + 24 = 144 answers.

* * * * *

Heh, heh... four different answers, I see. Have fun figuring out who is right. :)

Answer 2

1*1*12=12
1*2*6 = 12
1*3*4 = 12
2*2*3 =12

For each of the solutions above there are 8 ways in which a postive number can be replaced by negative so that the equation still holds. This means there are 32 soultions

Answer 3

12 = 12 x 1 x1
12 = 2 x 6 x 1
12 = 2 x 2 x3
12 = 4 x 3 x 1
or the absolute values ot their negatives

Answer 4

x can be +/-1 y=+/-1 and z= +/-12 for 8 different solutions there are 16 more solutions if x =+/- 12 and y=+/- 12 for a total of 24 solutions Another 24 solutions for 1,2,6
Another 24 for 1,3,4. And another 24 solutions for 2,2,3.
So total is 96 solutions.

Answer 5

12 factored,
(1,1,12)
(1,2,6)
(1,3,4)
(2,2,3)==>4 sets in all

but thats when theyre all counted as +ve no. we could hv them all +ve, only 1 -ve, 2 -ve, or all -ve.
all +ve =all -ve =1 way
1 different from the other 2 = 3 ways
so now we hv 8 ways, for each set.

now, to pair them out, we hv to divide the 4 sets.
(1,2,6) and (1,3,4) hv no repetition within set, while (1,1,12) and (2,2,3) do.

for the 1st 2, they could be paired in 6*8 ways, for each.that give us 2*48=96.

for the latter 2, they have 6 ways when the repetition hv different signs(one is +ve, and the other -ve) and only 3 for the same signs. from all 8 ways; [+++] and [---] make 2 same signs, and [++-] plus[--+] making it 4. so for (1,1,12) and (2,2,3), we hv 2*(6*4+3*4)=72

sum those up, we hv 96+72=168