Arranje os dígitos de 1 a 9 em uma ordem tal que os primeiros dois dígitos formem um número que seja um múltiplo de 2, os primeiros três dígitos formem um número que seja múltiplo de 3, e assim por diante. Exemplo: 123456789 não funciona, porque 12 é divisível por 2 e 123 é divisível por 3, mas 1234 não é divisível por 4.
(Do livro "Things to Make and Do in the Fourth Dimension", de Matt Parker)
(Do livro "Things to Make and Do in the Fourth Dimension", de Matt Parker)
Arrange the digits from 1 to 9 in such an order that the first two digits form a number that is a multiple of 2, the first three digits form a number that is a multiple of 3, and so on. Example: 123456789 doesn't work, because 12 is divisible by 2 and 123 is divisible by 3, but 1234 isn't divisible by 4.
(From Matt Parker's book "Things to Make and Do in the Fourth Dimension")
(From Matt Parker's book "Things to Make and Do in the Fourth Dimension")
PCFilho
Observação: há apenas uma solução possível para esse problema.
ResponderExcluirThe order is 381654729.
ResponderExcluir38/2=19
381/3=127
3816/4=954
38165/5=7633
381654/6=63609
3816547/7=545221
38165472/8=4770684
381654729/9=42406081
Well done, Jake!!
ExcluirThe number is 381,654,729.
I found it after observing that:
ResponderExcluir- the 5th digit has to be 5, as the number must be divisible by 5.
- the even digits have to be even, as the numbers must be divisible by 2.
- as a consequence of the previous statement, the odd digits have to be odd.
- the 3rd and 4th digits must be 12, 16, 32, 36, 72, 76, 92 or 96, as they have to be odd-even, not 5, and divisible by 4.
- in order to fulfill all the previous conditions and divisibility by 8, the 6th, 7th and 8th digits must be 216, 296, 416, 432, 472, 496, 632, 672, 816, 832, 872 or 896.
- combining the previous two statements and the statement about the 5th digit, the only possibilities from 3rd to 8th digits are: 125496, 125896, 165432, 165472, 165832, 165872, 325416, 325496, 325816, 325896, 365472, 365872, 725416, 725496, 725816, 725896, 765432, 765832, 925416, 925816, 965432, 965472, 965832 or 965872.
- in all the possibilities of the previous statement, 2 and 6 are present. As the 2nd digit is the only even digit outside that range (3rd-8th), it must be 4 or 8.
- combining informations from the previous statements, and applying the rule for divisibility by 3, the only possibilities for the first 3 digits are: 147, 183, 189, 381, 387, 741, 783, 789, 981 or 987.
(from the initial 9! = 362880 candidates, now we have only 20 possibilities.)
- combining the previous statements, the remaining candidates are:
147258963 (fails /7)
147658329 (fails /6)
183254967 (fails /6)
183654729 (fails /7)
189654327 (fails /7)
189654723 (fails /7)
381254967 (fails /6)
381654729 (our answer)
387254169 (fails /6)
387254961 (fails /6)
741258963 (fails /7)
741658329 (fails /6)
783254169 (fails /6)
783254961 (fails /6)
789254163 (fails /6)
789654321 (fails /7)
981654327 (fails /7)
981654723 (fails /7)
987254163 (fails /6)
987654321 (fails /7)
- as the other 19 numbers fail the tests of divisibility by 6 or 7, our answer is 381654729.
Pelos meus testes o numero 123 648 759 tambem funciona. O que estou fazendo errado para dar certo?
ResponderExcluirJa achei kkkkk falha na interpretação. Estava dividindo o sempre pelo ultimo numero. O que nao é o objetivo.
ExcluirIsso. No número que vc citou, 12364 não é divisível por 5.
Excluir