Question 29
For a positive integer $k$, we write
$$(1+x)(1+2x)(1+3x)\cdots(1+kx)=a_0+a_1x+a_2x^2+\cdots+a_kx^k,$$
where $a_0,a_1,\ldots,a_k$ are the coefficients of the polynomial. Find the smallest possible value of $k$ if $a_0+a_1+a_2+\cdots+a_{k-1}$ is divisible by $2005$.
$$(1+x)(1+2x)(1+3x)\cdots(1+kx)=a_0+a_1x+a_2x^2+\cdots+a_kx^k,$$
where $a_0,a_1,\ldots,a_k$ are the coefficients of the polynomial. Find the smallest possible value of $k$ if $a_0+a_1+a_2+\cdots+a_{k-1}$ is divisible by $2005$.