[jay_shark]

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

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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 .