— Burt Kaliski Jr. (@modulomathy) June 24, 2012
A four-digit representation base B covers numbers between B3 and B4-1. (As usual, it is assumed that the leading digit is at least 1.) Therefore, it’s sufficient to know which values of B satisfy both
B3 ≤ 175
B4-1 ≥ 175 .
The first condition is met by any base that is less than or equal to the cube root of 175, giving an integer upper bound of 5. The second, by any base that is greater than or equal to the fourth root of 175, giving an integer lower bound of 4. So the only solutions are base 4 and 5.
The base-4 representation of 175, to check, is:
175 = 2233 base 4
The base-3 and base-6 representations, to double-check are:
175 = 20111 base 3 — 5 digits
175 = 451 base 6 — 3 digits