- #1

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Find

- The numbers ##a_1,~a_2,~a_3,~a_4,~a_5,~a_6,~a_7,~a_8,~a_9,~a_{10}## are all distinct

- 1 divides ##a_1##

- 2 divides ##a_1a_2##

- 3 divides ##a_1a_2a_3##

- 4 divides ##a_1a_2a_3a_4##

- 5 divides ##a_1a_2a_3a_4a_5##

- 6 divides ##a_1a_2a_3a_4a_5a_6##

- 7 divides ##a_1a_2a_3a_4a_5a_6a_7##

- 8 divides ##a_1a_2a_3a_4a_5a_6a_7a_8##

- 9 divides ##a_1a_2a_3a_4a_5a_6a_7a_8a_9##

- 10 divides ##a_1a_2a_3a_4a_5a_6a_7a_8a_9a_{10}##

If you find this too easy: generalize this to to other bases.

**all**10-digit numbers ##a_1a_2a_3a_4a_5a_6a_7a_8a_9a_{10}## (for example, if ##a_1= 0##, ##a_2## = 1, ##a_3 = 2## and so on, we get the number 0123456789), such that all the following hold:- The numbers ##a_1,~a_2,~a_3,~a_4,~a_5,~a_6,~a_7,~a_8,~a_9,~a_{10}## are all distinct

- 1 divides ##a_1##

- 2 divides ##a_1a_2##

- 3 divides ##a_1a_2a_3##

- 4 divides ##a_1a_2a_3a_4##

- 5 divides ##a_1a_2a_3a_4a_5##

- 6 divides ##a_1a_2a_3a_4a_5a_6##

- 7 divides ##a_1a_2a_3a_4a_5a_6a_7##

- 8 divides ##a_1a_2a_3a_4a_5a_6a_7a_8##

- 9 divides ##a_1a_2a_3a_4a_5a_6a_7a_8a_9##

- 10 divides ##a_1a_2a_3a_4a_5a_6a_7a_8a_9a_{10}##

If you find this too easy: generalize this to to other bases.

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