The total number of positive integral solutions x , y , z such that x y z = 24 is :

Step-by-step solution

We have, x y z = 24 ⇒ x y z = 2 3 × 3 1 Let x = 2 α 1 × 3 β 1 y = 2 α 2 × 3 β 2 z = 2 α 3 × 3 β 2 where, α 1 + α 2 + α 3 = 3       . . . i β 1 + β 2 + β 3 = 1       . . . . ii Number of non-negative integral solution for i = C 2 5 = 10 Number of non-negative integral solution for ii = C 2 3 = 3 Total number of positive integral solutions x , y , z such that x y z = 24 is = 10 × 3 = 30