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Getting the bases with aabbcc (Posted on 2008-10-07) Difficulty: 3 of 5
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution only one | Comment 1 of 5

 

ANSWER:

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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