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Base Baffle (Posted on 2016-05-02) Difficulty: 3 of 5
A positive integer is expressible as 3X1 in base B and 1X3 in base B+6, where X is a base B (and also base B+6) digit.

Find all values of B and X.

Prove that no further solution is possible.

*** Of course, B is a positive integer.

No Solution Yet Submitted by K Sengupta    
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Solution solution Comment 2 of 2 |
3*b^2 + x*b + 1 = (b+6)^2 + x*(b+6) + 3
                =  b^2 + 12*b + 36 + x*b + 6*x + 3
                
2*b^2 - 12*b - 6*x - 38 = 0   

b^2 - 6*b - 3*x - 19 = 0

b = (6 + sqrt(36 + 12*x + 76)) / 2

b = 3 + sqrt(9 + 3*x +19) = 3 + sqrt(28 + 3*x)

x         b
7         10
12        11
24        13
...

All except the first are invalid as the digit x is larger than the base.

(Table function of graphing calculator app equivalent used; only integral b shown.)

Edited on May 2, 2016, 2:36 pm
  Posted by Charlie on 2016-05-02 14:35:06

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