Problem of the Fortnight #8

Alice and Bob live in the small principality of Binumeria, where everyone speaks only English and always tells the truth.  But, when it comes to numbers, two number systems are in common use.  By long tradition, natives never switch bases in mid-sentence.  Suppose Alice says, “I’m a firm believer in base 10 and speak only in that base.  The country has 26 people who use my base, and only 22 people speak base-14”.  Bob then adds, “Of the 25 residents, 13 are numerically bilingual and 1 is numerically illiterate.”  How many people live in Binumeria?

Solutions are due by 4pm on Friday, January 28^th in Goodrich 108

Problem courtesy of S. Forman