Divisor = Dividend $\times$ Quotient + Remainder So, $1186$ = Dividend $\times$ Natural number + $31$ Or Dividend $\times$ Natural number = $1186-31=1155$ Hence, the required natural number must be a factor of $1155$.

Now, $1155 = 1 \times 3 \times 5 \times 7 \times 11$ When $1186$ is divided by the required natural number, we get $31$ as remainder. Thus, the required natural number must be greater than $31$. Factors of $1155$ which are greater than $31$ are: $33$, $35$, $55$, $77$, $105$, $165$, $231$, $385$, and $1155$. Thus, there are 9 natural numbers which give a remainder of $31$ when $1186$ is divided by them.

Hence, option (D) is correct.