Divisibility 1,2,3,4,5,6,7,8,9,&10 Tried:
Seems the ten-digit number ends with $240$ or $640$ or $840$ (Is not true, there are more ways the number could end)
$8325971640,$
$8365971240,$
$8317956240,$
$8291357640,$
$8325971640,$
$8235971640,$
$1357689240,$
$1283579640,$
$1783659240,$
$1563729840,$
$1763529840,$
$1653729840,$
$7165239840,$
$7195236840,$
$2165937840,$
$9283579640$
 A: Suppose the number is of form $N=jihgfedcba$. We may write:
$a+b+c+d+e+f+g+h+i+j+(10-1)b+(10^2-1)c+(10^3-1)d+(10^4-1)e+(10^5-1)f+(10^6-1)g+(10^7-1)h+(10^8-1)i+(10^9-1)j$A: We just need a number divisible by $5,7,8,9,11$ and everything else is automatic. Divisibility by $9$ isn't a concern, as the digits already sum to $45$ and $9\mid45$. The number must end with $0$ since it's even and divisible by $5$. The last three digits must be divisible by $8$, so they're some multiple of $040$ (of course $040$ isn't actually a valid candidate, since it repeats $0$ twice). Divisibility by $11$ means the alternating sum of the digits must be a multiple of $11$. Divisibility by $7$ means the first $9$ digits is a multiple of $7$. Now, I think the most efficient method is to write a program taking into account all of these parameters to find a numerical answer.
A: COMMENT.- It is clear that the required number, $N$, must be a multiple of $2^3\cdot3^2\cdot5\cdot7\cdot11=27720$ so we must have $N=27720x$ where $x$ is such that   $N$  have ten (distinct) digits. It follows after a calculation$ $ that if there is solution then $x$ is an integer such that$$36076\le x\le360750$$ In other words, $x$ is a number belonging to a set of $324675$ integers. 
A: The number which are divisible by $8$ is also divisible by $2$ and $4$.
The number which are divisible by $9$ is also divisible by $3$.
The number of which is divisible by $6$ is also divisible by $2$ and $3$.
The number which are divisible by $10$ is also divisible by $2$ and $5$.
Also also the number we expect is divisible by $11$ and $7$.
So the number is in the form $=P×2^{3i}×3^{2j}×5^k×7^m×11^n$, 
Where $i$, $j$, $k$, $m$, & $n$ are positive any positive integer and $P$ is any positive integer integer.Using this condition we will produce a required ten digit number. 
