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# Find the number

8,127 Recognized Expert Expert
A ten-digit number contains every digit from 0 to 9. The digits are
arranged so that the number formed by the first two digits, reading from
left to right, is divisible by 2, the number formed by the first 3 digits
is divisible by 3, the first 4 by 4, and so on until the whole number is
divisible by 10.

can you please find the number ...!
Jul 16 '07 #1
5 2564
Stang02GT
1,208 Recognized Expert Top Contributor
im not gonna lie this ones tricky :}
Jul 16 '07 #2
jkmyoung
2,057 Recognized Expert Top Contributor
Let the 10 digits of the number be a-j, eg:
abcdefghij
abcdefghi0 div by 10
abcd5fghi0 div by 5

div by 2
even digits must be in 2, 4, 6, 8, 0
odd digits must be in 1, 3, 5, 7, 9.

div by 3
a+b+c mod 3 = 0
d+e+f mod 3 = 0
g+h+i mod 3 = 0

div by 4
Since c is odd, d must be 2 or 6.
similarly, since g is odd, h must be 2 or 6.
-> b and f are 4 and 8 in some order.

div by 8
Since f is 4 or 8, gh mod 8 = 0. Possibilities for gh:
32, 72, 16, 96

combined with div by 9 (3), the possibilities for ghi:
321, 327, 723, 729, 963,

d + 5 + f (mod 3) = 0
d + f = 1 (mod 3)
possibilities for def, given that f = 4 or 8 and e = 5
654, 258

do div by 7 seperately. So combining the options, and then taking mod of the first 7 digits we have:

987654321 => 5 mod 7
789654321 => 4 mod 7
981654327 => 2 mod 7
189654327 => 5 mod 7
981654723 => 6 mod 7
189654723 => 2 mod 7
183654729 => 6 mod 7
381654729 => 0 mod 7
147258963 => 6 mod 7
741258963 => 2 mod 7

3816547290
Jul 16 '07 #3
Stang02GT
1,208 Recognized Expert Top Contributor
wow....very impressive....i dunno if i ever would have figured that out
Jul 16 '07 #4
debasisdas
8,127 Recognized Expert Expert
3816547290
thanx for the explanation
Jul 17 '07 #5
Person
67 New Member

Sorry, didn't understand problem (don't know how to delete a post).
Jul 17 '07 #6