The number of the positive integral solutions x , y , z of the equation x y z = 24 is t , then the number of all possible factors of t is

Step-by-step solution

24 = 1.1 . 24 = 1.2 . 12 = 1.3 . 8 = 1.4 . 6 = 2.2 . 6 = 2.3 . 4 Each of 1,2 , 12 ,   1,3 , 8 ,   1,4 , 6 ,   2,3 , 4 can be permuted in 3 ! ways Each of 1,1 , 24 , 2,2 , 6 can be permuted in 3 ways The desired number of ways is 4 × 6 + 2 × 3 = 24 + 6 = 30 ⇒ t = 30 = 2 × 3 × 5 Number of factors of 30 = 2 × 2 × 2 = 8