Show that exists some positive integer $N$ such that among $1,2,...,N$, there are at least $0.99N$ good numbers.

Given a positive integer $k$, call $n$ good, if among $$\binom{n}{0},\binom{n}{1},\binom{n}{2},...,\binom{n}{n}$$ at least $0.99n$ of them are divisible by $k$. Show that exists some positive integer $N$ such that among $1,2,...,N$, there are at least $0.99N$ good numbers.

some try: for all prime $p$ and nonnegative integers $m,n$,
$$\nu_p \binom{m+n}{n} = \frac{s_p(m)+s_p(n)-s_p(m+n)}{p-1}$$
which is equal to the number of carries when adding $m$ and $n$ in mod $p$.