Answer: Option B
:
B
Minimum occurs when the numbers are 6, 6, 6, 2, 2, 1 (LCM = 6). Maximum occurs when the numbers are 7, 5, 4, 3, 2, 2 (LCM = 420).
To achieve minimum LCM, we have to search for maximum number of GCDs among the ${}^6{C}_2$ pairs. To look for maximum LCM, we need the numbers as co-prime.
Thus, M = 414. Now, 414 can be written as $2\times 3^2\times 23⟹$ M has $2\times 3\times 2$ = 12 divisors.
$M^2$ = $2^2\times 3^4\times {23}^2⟹$ $M^2$ has $3\times 5\times 3$ = 45 divisors.
Hence, option (b) (45-12 = 33) is the correct answer.