Stealing Chocolate

Ingyu loves chocolate, and he plans to steal Yeonjin's chocolate one piece at a time. Yeonjin has a chocolate of size $N \times M$. Ingyu will steal one piece from it, leaving $N \times M - 1$ pieces of chocolate. Stealing a piece changes the shape, so it would reveal that he stole it. Therefore Ingyu will rearrange the remaining chocolate into a rectangle.

The thing to watch is that the difference between the width and height of the new chocolate, $\texttt{D}\_{\texttt{new}}$, must not differ much from the difference between the width and height of the original chocolate, $\texttt{D}\_{\texttt{old}}$, or it will be obvious that he stole a piece. So Ingyu will keep $|\texttt{D}\_{\texttt{new}} - \texttt{D}\_{\texttt{old}}|$ at most $K$ and eat as many pieces as possible.

Consider the figure below as an example.

When $K=5$, there are $4 \times 4=16$ pieces of chocolate. The original chocolate has equal width and height, so $\texttt{D}\_{\texttt{old}}$ is $0$.

If Ingyu eats one piece and rearranges the rest into a $3 \times 5$ shape, then $\texttt{D}\_{\texttt{new}} = |3 - 5| = 2 \le \texttt{D}\_{\texttt{old}} + K = 5$, so he can eat one piece.

If he eats one more piece here, he can rearrange the rest into a $2 \times 7$ shape. Then $\texttt{D}\_{\texttt{new}} = |2-7| = 5 \le \texttt{D}\_{\texttt{old}} + K = 5$, so he can eat one more piece.

If he eats one more piece here, the only possible rearrangement is a $1 \times 13$ shape. Then $\texttt{D}\_{\texttt{new}} = |1-13| = 12 > \texttt{D}\_{\texttt{old}} + K = 5$, which does not satisfy the rule, so he cannot eat a third piece.

Here is another example.

When $K=4$, there are $4 \times 5 = 20$ pieces of chocolate. From the original chocolate, $\texttt{D}\_{\texttt{old}}$ is $1$.

If Ingyu eats one piece, he must arrange $19$ pieces into a rectangle, and the smallest $|\texttt{D}\_{\texttt{new}} - \texttt{D}\_{\texttt{old}}|$ is $18$, which is greater than $K$, so Ingyu cannot eat any piece. If Ingyu eats $2$ pieces and then arranges the rest into a $3 \times 6$ shape, $|\texttt{D}\_{\texttt{new}} - \texttt{D}\_{\texttt{old}}|$ is $2$, which is less than $K$, but since he can eat only one piece at a time, he cannot reach that state.

Given $N,\ M,\ K$, find the number of Yeonjin's chocolates that Ingyu can secretly eat.

However, if Ingyu eats all the chocolate, Yeonjin will notice, so at least $1$ piece of chocolate must remain.

The first line contains three integers $N,\ M,\ K$ separated by spaces. ($1 \le N,\ M \le 10\,000$, $0 \le K \le N \times M$)

Print the number of chocolates that Ingyu can secretly eat.