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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.

  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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(2): only oneAdy TZIDON2008-10-07 14:17:30
Some Thoughtsre(2): only oneAdy TZIDON2008-10-07 14:03:49
re: only oneCharlie2008-10-07 13:13:20
SolutionsolutionsCharlie2008-10-07 13:03:41
Solutiononly oneAdy TZIDON2008-10-07 11:49:27
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