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

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Prove that the term *n* = 120 in This sequence is A072507 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

is 0:

If 24*k 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!