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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?

 Jun 24, 2020
 #1
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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.

Now

(# of residents) = (# of M-users) + (# of (M+4)-users) – (# of users of both) + (# of users of neither).

Therefore

2(M+4) + 5 = (2M+6) + (2M+2) – ((M+4) + 3) + 1,

so 11 = M, whereby 2(M+4) + 5 =

@jibz yahoo,

This is the answer please figure out the rest, 

 Jun 24, 2020
 #2
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Thankyou so much :)

ItzMe  Jun 24, 2020

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