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 Productive Numbers (Posted on 2013-05-26)
Find all possible 9-digit numbers using precisely nine distinct digits from 0 to 9 such that each pair of consecutive digits is the product of two single-digit numbers.

For example, 2 and 1 can appear next to each other since 3*7=21. However, 2 and 6 cannot appear next to each other since 26 is the product only of 2 and 13 (and 1 and 26), and each pair of factors contains a number that is not a single-digit number.

*** None of the 9-digit numbers can contain any leading zero.

 No Solution Yet Submitted by K Sengupta No Rating

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 10-digit decimal numbers Comment 6 of 6 |

Charlie's long-list shows that there are 229 non-leading
zero numbers where any consecutive pair of digits treated as a two-digit, zero-leading permitted number can have single-digit factors.

Extending the number to include one of each of the ten decimal digits, only 25 such numbers exist.

3072815649   5*6 1*7 8*9 4*7 9*9 3*5 7*8 8*8 7*7
5630728149   7*8 7*9 5*6 1*7 8*9 4*7 9*9 2*7 7*7
5630728149   7*8 7*9 5*6 1*7 8*9 4*7 9*9 2*7 7*7
6307281549   7*9 5*6 1*7 8*9 4*7 9*9 3*5 6*9 7*7
6307281549   7*9 5*6 1*7 8*9 4*7 9*9 3*5 6*9 7*7
7208163549   8*9 4*5 1*8 9*9 2*8 7*9 5*7 6*9 7*7
7208163549   8*9 4*5 1*8 9*9 2*8 7*9 5*7 6*9 7*7
7248156309   8*9 4*6 6*8 9*9 3*5 7*8 7*9 5*6 1*9
7254816309   8*9 5*5 6*9 6*8 9*9 2*8 7*9 5*6 1*9
7256308149   8*9 5*5 7*8 7*9 5*6 1*8 9*9 2*7 7*7
7256308149   8*9 5*5 7*8 7*9 5*6 1*8 9*9 2*7 7*7
7281035649   8*9 4*7 9*9 2*5 1*3 5*7 7*8 8*8 7*7
7281035649   8*9 4*7 9*9 2*5 1*3 5*7 7*8 8*8 7*7
7281063549   8*9 4*7 9*9 2*5 1*6 7*9 5*7 6*9 7*7
7281063549   8*9 4*7 9*9 2*5 1*6 7*9 5*7 6*9 7*7
7281456309   8*9 4*7 9*9 2*7 5*9 7*8 7*9 5*6 1*9
7281563049   8*9 4*7 9*9 3*5 7*8 7*9 5*6 1*4 7*7
7281563049   8*9 4*7 9*9 3*5 7*8 7*9 5*6 1*4 7*7
7281630549   8*9 4*7 9*9 2*8 7*9 5*6 1*5 6*9 7*7
7281630549   8*9 4*7 9*9 2*8 7*9 5*6 1*5 6*9 7*7
7281635409   8*9 4*7 9*9 2*8 7*9 5*7 6*9 5*8 1*9
8156307249   9*9 3*5 7*8 7*9 5*6 1*7 8*9 4*6 7*7
8156307249   9*9 3*5 7*8 7*9 5*6 1*7 8*9 4*6 7*7
8163072549   9*9 2*8 7*9 5*6 1*7 8*9 5*5 6*9 7*7
8163072549   9*9 2*8 7*9 5*6 1*7 8*9 5*5 6*9 7*7

 Posted by Dej Mar on 2013-05-27 01:24:55

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