What the difference between the smallest two numbers from these numbers? There are infinitely many integers $n$ bigger than $1$, such that if we divide $n$ by any integer $k$ where $2\leq k\leq 11$, the remainder is equal  to $1$.
What the difference between the smallest two such integers?
Help guys please. this is my try, the smallest number will be $lcm+1$ $$= 2^3 * 3^2 * 5*7*11+1$$ and the next number will be $$2*lcm+1$$
the difference between them $=icm=27720$

What you wan can be set up as:
$x\equiv1\bmod 2$
$x\equiv 1\bmod 3$
$x\equiv1\bmod 4$
$x\equiv 1\bmod 5$
$x\equiv1\bmod 6$
$x\equiv 1\bmod 7$
$x\equiv1\bmod 8$
$x\equiv 1\bmod 9$
$x\equiv1\bmod 10$
$x\equiv 1\bmod 11$
which is equivalent to
$x\equiv 1 \bmod 2^3$
$x\equiv 1 \bmod 3^2$
$x\equiv 1 \bmod 5$
$x\equiv 1 \bmod 7$
$x\equiv 1 \bmod 11$
which by CRT has unique solution $\bmod 2^3\cdot3^2\cdot 5\cdot 11$. The answer thus is $2^3\cdot3^2\cdot 5\cdot7 \cdot11$
