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Find number and base (Posted on 2016-03-17) Difficulty: 2 of 5
  

An integer n written in base b is 211 and 110
written in base b+2. Find n and b in base 10.   

No Solution Yet Submitted by Bractals    
Rating: 5.0000 (1 votes)

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Solution solution | Comment 1 of 6
2*b^2 + b + 1 = n

(b+2)^2 + b + 2 = n

b^2 + 4*b + 4 + b + 2 = n

b^2 + 5*b + 6 = n

Subtracting the first equation from twice this gives

9*b + 11 = n

So 

2*b^2 + b + 1 = 9*b + 11

2*b^2 - 8*b - 10 = 0

b^2 - 4*b - 5 = 0

b = (4 + sqrt(16 + 20)) / 2
  = 2 + sqrt(9)
  = 5
  
n = 9*5 + 11 = 56  

  Posted by Charlie on 2016-03-17 10:41:11
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