# Number System Conversion

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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.

• 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? – String Mar 30 '15 at 9:02
• 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. – iOSAndroidWindowsMobileAppsDev Mar 30 '15 at 9:04
• 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. – String Mar 30 '15 at 9:07
• I don't know why you think the question requires a fractional base system. Why can't it be about base 10? – Gerry Myerson Mar 30 '15 at 9:08
• 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. – kate Mar 31 '15 at 1:18