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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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 solutions | Comment 2 of 5 |

The following program does not check that a, b and c are all different, but no cases of duplication were found among them:

t = 2
DO
n = t * t + t + 1
n6\$ = ""
DO
r = n MOD 6
n = n \ 6
n6\$ = LTRIM\$(STR\$(r)) + n6\$
LOOP UNTIL n = 0
IF LEN(n6\$) = 6 THEN
IF MID\$(n6\$, 1, 1) = MID\$(n6\$, 2, 1) THEN
IF MID\$(n6\$, 3, 1) = MID\$(n6\$, 4, 1) THEN
IF MID\$(n6\$, 5, 1) = MID\$(n6\$, 6, 1) THEN
PRINT t, n6\$
END IF
END IF
END IF
END IF
t = t + 1
LOOP UNTIL LEN(n6\$) > 6

resulting in:

`base     base-6 representation  100          114433 137          223311`

 Posted by Charlie on 2008-10-07 13:03:41

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