For a positive integer $k,$ we write $\left(1+x\right)\left(1+2x\right)\left(1+3x\right)\cdots \left(1+kx\right)=a_0+a_{1}x+a_2{x}^2+\cdots +a_k{x}^k$ where $a_0, a_1, \cdots , a_k$ are the coefficients of the polynomial. Find the sum of all the digits of the smallest possible value of $k$ if $a_0+a_1+a_2+\cdots +a_{k-1}$ is divisible by $2005$

The smallest three-digit number $n$ such that if the three digits are $a, b$ and $c,$ then $n=a+b+c+ab+bc+ac+abc$

Evaluate $\frac{n+1}{5}$

Let the sum of the digits of the given product is written as $A 8 B 6 C.$ Find $A+B+C$.

 $\underset{2007 \text{times}}{\underbrace{\left(111111111\dots 111\right)}}\times 2007$

Let the number of distinct integers among the following numbers is $N.$ Find the sum of all digits of $N.$

$\left[\frac{1^2}{2007}\right],\left[\frac{2^2}{2007}\right],\dots ,\left[\frac{{2007}^2}{2007}\right]$, where, $\left[·\right]$ is the greatest integer function.