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 Back to the base (Posted on 2011-03-30)
A certain 10-base integer with distinct digits can be converted to base b by merely reversing its digits.
Find the lowest value of b.

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 computer solution (spoiler) | Comment 1 of 3

DEFDBL A-Z
CLS
FOR b = 3 TO 9
FOR d1 = 0 TO b - 1
FOR d2 = 0 TO b - 1
IF d1 * 10 + d2 = d2 * b + d1 AND d1 + d2 > 0 THEN PRINT b, d1; d2
NEXT d2
NEXT d1
NEXT b

finds for 2-digit numbers with bases under 10:

` b   decimal digits to be reversed 4             1  3 7             2  3 7             4  6`

So the smallest b seems to be 4, where 13 decimal can be represented as 31.

Going to 3-digit numbers would not seem to be able to lower b to 3, as the highest 3-digit base-3 number would be represented as 222 and be the equivalent to 18+6+2=26 decimal, which is only a 2-digit number.

 Posted by Charlie on 2011-03-30 13:02:15

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