Question 522677
In a numeration system with a positive integral base, the numerals 104 and 241
represent the degree –measures of a pair of supplementary angles. What is the
base of this numeration system?
<pre>
Let the base be b. Then

104<sub>b</sub> = 1·bē + 0·b + 4 = bē + 4

241<sub>b</sub> = 2·bē + 4·b + 1 = 2bē + 4b + 1

Since they are the measures of supplementary angles,

(bē + 4) + (2bē + 4b + 1) = 180

    bē + 4 + 2bē + 4b + 1 = 180
    
             3bē + 4b + 5 = 180 
            
           3bē + 4b - 175 = 0

         (b - 7)(3b + 25) = 0

      b - 7 = 0       3b + 25 = 0
          b = 7            3b = -25
                            b = {{{-25/3}}}

Ignore the negative fractional anwser.

Solution:  Base 7.

Checking:
104<sub>7</sub> = 1·7ē + 4 = 49 + 4 = 53°
241<sub>7</sub> = 2·7ē + 4·7 + 1 = 2(49) + 28 + 1 = 98 + 28 + 1 = 127°

and 53° + 127° = 180°

Edwin</pre>