In the small hamlet of Abaze, two base systems are in common use. Also, everyone speaks the truth, and no one changes which base they are using in mid-sentence. One resident said, "26 people use my base, base 10, and only 22 people speak base 14." Another said, "Of the 25 residents, 13 use both bases and 1 can't use either base." How many residents are there?
Let M and N be the base in which 1st and 2nd speaker talked, respectively. Then according to the 1st speaker
2M+6 residents use base M, and
2M+2 residents use base M+4,
which means that N = M+4. According to the second speaker
2(M+4) + 5 people reside in Abaze,
(M+4) + 3 people use both bases, and
1 resident uses neither base.
(# of residents) = (# of M-users) + (# of (M+4)-users) – (# of users of both) + (# of users of neither).
2(M+4) + 5 = (2M+6) + (2M+2) – ((M+4) + 3) + 1,
so 11 = M, whereby 2(M+4) + 5 =
This is the answer please figure out the rest,