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I have a paradox: EIGHTY is a six digit number with no repeating digits and no zeros. When divided by 19, 17, 13, 11, or H, the remainders are, respectively, 17, 13, 11, 7 and G.

TWENtY is (another) six digit number with no repeating digits and no zeros (and uses a different key to EIGHTY above). When divided by T, perfect square WE or perfect cube NtY, the remainder is zero.

Find EIGHTY TWENtY

My interpretation is: The question requires a fractional base system converion e.g. 20 converted to base 2 is and 0010100 and 20 converted to base 1.6 is approximately 1001001.2589 which is a six digit number but both have repeating digits and zeros.

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  • $\begingroup$ I am pretty sure that aplhabets are digits in the system too, so EIGHTY has nothing to do with $80$. So EIGHTY has digits E,I,G,H,T and Y - thus six digits, no repeating digits, right? $\endgroup$ – String Mar 30 '15 at 9:02
  • $\begingroup$ you cannot give a solution ad neglect the second most important part - i.e. When divided by 19, 17, 13, 11, or H, the remainders are, respectively, 17, 13, 11, 7 and G. $\endgroup$ – iOSAndroidWindowsMobileAppsDev Mar 30 '15 at 9:04
  • $\begingroup$ Sorry that I did not make that clear - I was merely making a comment about the interpreation of the question, not claiming to have a solution. $\endgroup$ – String Mar 30 '15 at 9:07
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    $\begingroup$ I don't know why you think the question requires a fractional base system. Why can't it be about base 10? $\endgroup$ – Gerry Myerson Mar 30 '15 at 9:08
  • $\begingroup$ As you commented below, this question was crossposted to Puzzling.SE, where it received a very thorough answer. I don't think it's useful to leave it posted here. $\endgroup$ – kate Mar 31 '15 at 1:18
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I could find TWENtY = 349125. Since WE is perfect square, it belongs to set: {25,36,49,49,64,81} and NtY is perfect cube, it belongs to set: {125,216,343,512,729}. Now, using the given information of no repeating digits, no zeros and the remainder, we can make combinations of WE and NtY to find TWENtY. I could not find EIGHTY, but I have a question here. Are the digits 'T','E','Y' in TWENtY the same as that in EIGHTY ?

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