4楼老猫
(谦虚使人进步,骄傲使人快乐。)
发表于 2009-2-8 22:11
只看此人
英文好的,可以看这个。也是网上搜来的。
RogerioAug27-04, 02:14 PM
Rogerio, could you please share your reasoning with us? Thank you.
OK, let's go!
(I'm gonna use the symbol '=' with the extra meaning 'belongs to')
The 10 digits number is divisible by 10 -> '0' is the last digit.
The 9 digits number is divisible by 9 -> as the first nine digits sum 45, don't care about the last digit.
The 5 digits number is divisible by 5 -> '0' is not available, so '5' is the last digit.
Our whole number is 'A B C D 5 E F G X 0' .
AB is divisible by 2 -> B is even
ABCD is divisible by 4 -> D is even , and so on.
B,D,E,G belong to [2,4,6,8]
A,C,F belong to [1,3,7,9]
C is odd and ABCD is div by 4 -> D = [2,6]
F is odd and ABCD5EFG is div by 8 -> G = [2,6]
So, D,G=[2,6] and B,E=[4,8]
ABC is div by 3, so A+B+C is div by 3
But ABCD5E is div by 6 and 3, too. So D+5+E is div by 3
Since ABC is div by 3, and B=[4,8]
if B=4 -> A,C=[1,7] So, E=8 and F=[3,9]. But D5E is div by 3 , and D=[2,6] , so D=2 . So G=6. But FG should be div by 8, so F=9 .
So, ABCD5EF = 1472589 or 7412589 which are not div by 7.
So B can't be '4' .
Then B=8 (and E=4).
E=4 and D5E is div by 3 -> D=6 . So G=2.
B=8 and ABC is div by 3 -> A,C=[1,3] or A,C=[7,9]
if A,C=[7,9] -> F=[1,3] but FG is div by 8 so F=3.
So ABCD5EF = 7896543 or 9876543 , which are not div by 7.
if A,C=[1,3] -> F=[7,9] but FG is div by 8 so F=7.
So ABCD5EF = 1836547 or 3816547 . And only 3816547 is div by 7.
So, the number is 3816547290 :-).