The problem is as follows:
Dividing the number of oranges by any odd digit and you end up with the next smallest even digit left over. What is the smallest number of oranges?
Let $N$ be the number of oranges. Then dividing by any odd number and setting it equal to the next smallest even number we obtain
$$\frac{N}{2n+1} = 2n \implies N = 2n(2n+1)$$
The smallest is when $n = 1$. So $N=6$
Apparently the answer is not 6, but 314. Here is the justification for that answer, which I would like an explanation of.
Numerically, for each odd digit $n$ we divide by, we have $n-1$ left over. So add an orange. Then the number is divisible by $3,5,7$ and $9$. [This I don't understand]. The case of 3 is covered by 9. So the number is divisible without remainder by $5,7$ and $9$. These are all mutually prime so the smallest such number above $0$ is $5.7.9 = 315$. One fewer would be $314$. [why?]
As an example, you are told that when you divide by $7$ you have $6$ left over. If you add an orange the number will be divisible by $7$. The same applies to $3,5,9$. This justifies the claim that one more than the number of oranges must be a multiple of $5,7, \text { and } 9$. $5 \cdot 7 \cdot 9=315$