There are no 120 consecutive numbers which all have exactly 120 divisors

Prove that the term $n=120$ in is $0$:

If $24k$ with $k$ coprime to $6$ has exactly $120$ divisors, than $k$ has exactly $15$ divisors, thus $k$ is a square number, thus k cannot be == 5, 7, 11 mod 12 (since 5, 7, 11 are not quadratic residues mod 12), thus a number == 120, 168, 264 mod 288 cannot have exactly 120 divisors (since such numbers can be written as 24*k with k coprime to 6 and k == 5, 7, 11 mod 12), thus if there are 120 consecutive integers with exactly 120 divisors, than the start number must be == 0, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287 mod 288, and hence == 0, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 mod 32, thus there are 4 consecutive multiples of 32 among these 120 integers, and one of these 4 numbers must be == 64 mod 128, thus the number of divisors of this number must be divisible by 7 and cannot be 120, which is a contradiction!