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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: (Hide) 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.

 Subject Author Date re(2): only one Ady TZIDON 2008-10-07 14:17:30 re(2): only one Ady TZIDON 2008-10-07 14:03:49 re: only one Charlie 2008-10-07 13:13:20 solutions Charlie 2008-10-07 13:03:41 only one Ady TZIDON 2008-10-07 11:49:27

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