— Burt Kaliski Jr. (@modulomathy) August 28, 2013
For clarity, we’ll assume that “two digits in increasing consecutive order” means that, like 0xef, there are exactly two digits in the representation of the number — not just that two adjacent digits have this order.
(Recall that the notation 0xef means that the digits base 16 are 14 and 15; e is the symbol for 14 and f is the symbol for 15 in hexadecimal.)
Given the constraint of a two-digit representation, the possible bases are limited to the range 16 to 239. Smaller bases would require at least three digits (because 239 is greater than 152); larger bases would require only one.
Now let B be the base, and let x be the first digit of the representation. If the two digits are in increasing consecutive order, then the second digit is x+1, which means that we have
xB + (x+1) = 239 .
Rearranging, we have
x(B+1) + 1 = 239
x(B+1) = 238,
which means that (B+1) is a divisor of 238. The divisors of 238 are 1, 2, 7, 14, 17, 34, 119, and 238. Within the range 16 to 239, the possible bases are therefore 16, 33, 118, and 237. A quick check of representations confirms this; 239 can be represented as:
- (14 15) base 16
- (7 8) base 33
- (2 3) base 118
- (1 2) base 239
For the reader: If we assumed instead that that the solution only required that two adjacent digits be in increasing consecutive order — not that there be exactly two digits in the representation — would there be any other bases that worked?