Don't really understand this question. If this is asking to find an $x$ for each mod then the answer would be just be $x+m \dots$. If this is asking to find an $x$ that satisfies all mods, then this cant be since every $x$ mod $1$ would be $0$ and no $x$ would satisfy for all. Any help?
Every integer $x$ is also congruent to $1$ modulo $1$. There is no smallest integer satisfying your conditions, but there is a smallest positive integer, namely $1$. For certainly every integer from $1$ to $10$ divides $1-1$.
To see that there is no smallest integer, let $m$ be the lcm of the numbers $1$ to $10$. Then the integers that satisfy your condition are precisely the integers of the form $1+km$. Take $k=-1$.
If you want the smallest integer $x$ greater than $1$, the answer is $1+m$. Note that $m=2^3\cdot 3^2\cdot 5\cdot 7=2520$.
As the remainder is same $=1,$ so $x-1$ needs to be divisible by $1,2,\cdots,9,10$
Now, if $1,2,\cdots,9,10$ divides $x-1,$ the later must be divisible by lcm $(1,2,\cdots,9,10)$