Find the smallest binomial distribution with two tied modes

The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success at any trial is $p$.

Denoting $P(X=k)$ by $p_k$, it is given that $\dfrac{p_k}{p_{k-1}}=\dfrac{(n-k+1)p}{k(1-p)}$, $k=1,2,\ldots,n$.

A distribution is said to be bimodal if it has two modes.

Find the least value of $n$, and the corresponding modes of $X$, given that $X$ is bimodal and that $p=\dfrac{18}{25}$.3 marks