How many sets of three distinct factors of t...

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How many sets of three distinct factors of the number N = 26 x 34 x 52 can be made such that the factors in each set have a highest common factor of 1 with respect to every other factor in that set?
• a)
236
• b)
360
• c)
104
• d)
380

N = 26 × 34 × 52
Case: [A]
when none of the elements is 1 ⇒ the three factors must be some powers of 2, 3, 5 respectively.
⇒ total number of such sets = 6 × 4 × 2 = 48
Case: [B]
when one of the elements is 1
⇒ the other two factors could be of the form (2a), (3b) − number of sets = 6 × 4 = 24
(26), (52) − number of sets = 6 × 2 = 12
(34),(52) − number of sets = 4 × 2 = 8
(26 × 34 × 52) − number of sets = 6 × 4 × 2 = 48
(26 × 52 × 34) − number of sets = 6 × 2 × 4 = 48
(34 × 52 × 26) − number of sets = 4 × 2 × 6 = 48
∴ total number of such sets = 188
combining both the cases, total sets possible
= 188 + 48 = 236

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