Question 1121449
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Without context, the problem PROBABLY is looking for a solution in  positive integers.  However, if (a,b,c) is a solution, then (-a,-b,-c) is also a solution.<br>
The problem seems to suggest finding the product xyz first and then using that to find x, y, and z separately.<br>
That can be done; but the arithmetic gets ugly.  It is far easier to find the values of x, y, and z and then find the value of the product xyz.<br>
The values of the separate variables can be found easily by trial and error, by looking at the possible factorizations of the three given products.<br>
For me, the first thing I see is 117 = 9*13.
Then, having one number that has a factor of 9, I look for another; 198 = 9*22.
And trying 13*22 for the third product gives me the right result, 286.<br>
So the solution in positive integers is 9, 13, and 22.<br>
Finding the product of all three numbers is then simple arithmetic.