#1
|
|||
|
|||
Russia National Contest Pr 1998-7
A 10-digit number is said to be "interesting" if its digits are all distinct and it's a multiple of 11111. How many "interesting" integers are there?
|
#2
|
|||
|
|||
Re: Russia National Contest Pr 1998-7
I just had a solution written out but somehow it erased .
In any case , I figured out that if the 10 digit integer is abcdefghij then it turns out that the solutions are a+f=9 , b+g=9 , c+h=9 , d+i=9 , e+j=9 as long as we take into account that the first digit cannot be a 0 . So a solution may be 4321056789 or 5678943210 ... It's relatively easy to count the total number of solutions given the constraints . |
#3
|
|||
|
|||
Re: Russia National Contest Pr 1998-7
[ QUOTE ]
It's relatively easy to count the total number of solutions given the constraints . [/ QUOTE ] Do tell us the answer. |
#4
|
|||
|
|||
Re: Russia National Contest Pr 1998-7
[ QUOTE ]
I just had a solution written out but somehow it erased . In any case , I figured out that if the 10 digit integer is abcdefghij then it turns out that the solutions are a+f=9 , b+g=9 , c+h=9 , d+i=9 , e+j=9 as long as we take into account that the first digit cannot be a 0 . So a solution may be 4321056789 or 5678943210 ... It's relatively easy to count the total number of solutions given the constraints . [/ QUOTE ] unless im doing something wrong it seems the answer is 10 nPr 10, or 3628800 |
#5
|
|||
|
|||
Re: Russia National Contest Pr 1998-7
[ QUOTE ]
unless im doing something wrong it seems the answer is 10 nPr 10, or 3628800 [/ QUOTE ] You're counting a lot of numbers not in the solution. Remember, they need to be a multiple of 11111. |
#6
|
|||
|
|||
Re: Russia National Contest Pr 1998-7
[ QUOTE ]
[ QUOTE ] unless im doing something wrong it seems the answer is 10 nPr 10, or 3628800 [/ QUOTE ] You're counting a lot of numbers not in the solution. Remember, they need to be a multiple of 11111. [/ QUOTE ] then is it 10^5, if there are 10 combinations of x+y=9, and 5 sets of pairs |
#7
|
|||
|
|||
Re: Russia National Contest Pr 1998-7
It's not 10^5, one of the things you need to remember is that the first digit can not be zero.
|
#8
|
|||
|
|||
Re: Russia National Contest Pr 1998-7
[ QUOTE ]
It's not 10^5, one of the things you need to remember is that the first digit can not be zero. [/ QUOTE ] so then what is the correct way of doing it, 10,001 seems too obvious |
#9
|
|||
|
|||
Re: Russia National Contest Pr 1998-7
There are 10^5 solutions to the 5 equations .
Now we subtract the number of ways a=0 and f=9 . This means there are 8 available numbers for the four equations and so we have 10^5 - 8^4 = 95,904 ================================================== ======= Here is the solution : The 10-digit number can be written as a*10^9 + b*10^8 + c*10^7 + ...+ i*10^1 +j*10^0 a*10^9 =a*10^4mod11111 b*10^8=b*10^3mod11111 c*10^7=c*10^2mod11111 d*10^6=d*10^1mod11111 e*10^5=e*10^0mod11111 So we require (a+f)*10^4 + (b+g)*10^3 + ... + (e+j)*10^0 =y*11111 for 1<=y<=16 , since the maximum value for a+f = 17 and the minimum value for a+f = 1 . It turns out that the only solution is when a+f=9 , in which case b+g=9 , c+h=9 etc . |
#10
|
|||
|
|||
Re: Russia National Contest Pr 1998-7
[ QUOTE ]
There are 10^5 solutions to the 5 equations . [/ QUOTE ] All digits are distinct, so there are way less than 10^5 solutions |
|
|