Diophantine drives me crazy !
Determine the number of integral solutions with distinct, of
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1 solution
Factorizing gives 12=1·2·2·3, and thus, for |x|,|y| and |z| to be distinct, the possibilities for the three absolute values |x|, |y|, |z| are (ignoring order) (1, 2, 6) and (1, 3, 4). Each of these can be ordered in six different ways, and then, for each of the six ways, we can have 0, 1, 2, or 3 of the integers positive, making a total of eight ways. Thus altogether there are 2 · 6 · 8 = 96 solutions to identify.