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Getting the bases with aabbcc (Posted on 2008-10-07 )

Determine all possible positive integer base(s) T such that:
(111)_{T} = (aabbcc)_{6} , where each of a, b and c denotes a different base 6 digit from 0 to 5 and, a is not zero .
Note : Try to solve this problem analytically, although computer program/ spreadsheet solutions are welcome.

Submitted by K Sengupta
Rating: 2.0000 (1 votes)
Solution:
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By the given problem:
T^2 + T + 1 = 7(1296a + 36b + c) ……..(i)
or, T^2 + T + 1 (mod 7) = 0
or, T^2 + T + 1 – 7(T-1) (mod 7) = 0
or, (T^2 - 6T + 8) (mod 7) = 0
or, (T-4)(T-2)(mod 7) = 0
or, T(mod 7) = 2, 4 ……(ii)
Again:
T^2 < (T^2 + T + 1)= (aabbcc)_6 < 6^6
or, T < 216 …….(iii)
Checking for positive integer values T < 216 that satisfy (i) and (ii), we observe that: only T = 100, 137 are valid, whereby: aabbcc = 114433, 223311.
Consequently, the required values of T are 100 or 137.

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