This page contains some examples to clarify the rules for the Remainders game.

Suppose you have discovered that your number has a remainder of $1$ when divided by $2$, and a remainder of $0$ when divided by $5$, as in the example below:

At this point, you know that the hidden number is both odd and a multiple of $5$. You might think of $15$ as a number that would fit with both of these criteria.

However, there are still other possibilities that the number could be, such as $5$, $25$ or even $95$. This means that you'd need to do some more divisions to work out which number it is.

Suppose you are in the situation below, where you have divided by $4$ and found that you are left with a remainder of $1$.

At this stage, you know your number is $1$ or $5$ or $9$ or..., so all the possibilities that you get are odd. Therefore, we already know that, if you choose to divide by $2$, the remainder will be $1$.

So, if you did divide by $2$ in this situation you would gain no extra information as to what the hidden number is, so you would score no points.