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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.

 See The Solution Submitted by K Sengupta Rating: 2.0000 (1 votes)

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 only one | Comment 1 of 5

T=100  AND THE NUMBER BASE 6 IS 114433

SOLUTION:

N=T^2+T+1  is a multiple of 7  ( in base 6 it is represented by 11)

therefore t is 2 mod 7    i.e. 2,9,16,.....

Since  1296 <N/7<5*(1296+36+1)

ONLY FEW CANDIDATE SOLUTION EXIST

MANUAL CHECK LEAVES ONLY ONE:

T=100

N( BASE  10)=10101

N/7( BASE  10)=10101/7=1443=  10403( BASE 6)

AND THE NUMBER N  IN BASE  6   = 114433

Edited on October 7, 2008, 11:56 am
 Posted by Ady TZIDON on 2008-10-07 11:49:27

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