SOLUTION: 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

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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?
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
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?
Let the base be b. Then

104b = 1·b² + 0·b + 4 = b² + 4

241b = 2·b² + 4·b + 1 = 2b² + 4b + 1

Since they are the measures of supplementary angles,

(b² + 4) + (2b² + 4b + 1) = 180

    b² + 4 + 2b² + 4b + 1 = 180
    
             3b² + 4b + 5 = 180 
            
           3b² + 4b - 175 = 0

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

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

Ignore the negative fractional anwser.

Solution:  Base 7.

Checking:
1047 = 1·7² + 4 = 49 + 4 = 53°
2417 = 2·7² + 4·7 + 1 = 2(49) + 28 + 1 = 98 + 28 + 1 = 127°

and 53° + 127° = 180°

Edwin
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