Diverse Matrix
Description
Let $a$ be a matrix of size $r \times c$ containing positive integers, not necessarily distinct. Rows of the matrix are numbered from $1$ to $r$, columns are numbered from $1$ to $c$. We can construct an array $b$ consisting of $r + c$ integers as follows: for each $i \in [1, r]$, let $b_i$ be the greatest common divisor of integers in the $i$-th row, and for each $j \in [1, c]$ let $b_{r+j}$ be the greatest common divisor of integers in the $j$-th column.
We call the matrix diverse if all $r + c$ numbers $b_k$ ($k \in [1, r + c]$) are pairwise distinct.
The magnitude of a matrix equals to the maximum of $b_k$.
For example, suppose we have the following matrix:
We construct the array $b$:
- $b_1$ is the greatest common divisor of $2$, $9$, and $7$, that is $1$;
- $b_2$ is the greatest common divisor of $4$, $144$, and $84$, that is $4$;
- $b_3$ is the greatest common divisor of $2$ and $4$, that is $2$;
- $b_4$ is the greatest common divisor of $9$ and $144$, that is $9$;
- $b_5$ is the greatest common divisor of $7$ and $84$, that is $7$.
So $b = [1, 4, 2, 9, 7]$. All values in this array are distinct, so the matrix is diverse. The magnitude is equal to $9$.
For a given $r$ and $c$, find a diverse matrix that minimises the magnitude. If there are multiple solutions, you may output any of them. If there are no solutions, output a single integer $0$.
The only line in the input contains two space separated integers $r$ and $c$ ($1 \leq r,c \leq 500$) — the number of rows and the number of columns of the matrix to be found.
If there is no solution, output a single integer $0$.
Otherwise, output $r$ rows. The $i$-th of them should contain $c$ space-separated integers, the $j$-th of which is $a_{i,j}$ — the positive integer in the $i$-th row and $j$-th column of a diverse matrix minimizing the magnitude.
Furthermore, it must hold that $1 \leq a_{i,j} \leq 10^9$. It can be shown that if a solution exists, there is also a solution with this additional constraint (still having minimum possible magnitude).
Input
The only line in the input contains two space separated integers $r$ and $c$ ($1 \leq r,c \leq 500$) — the number of rows and the number of columns of the matrix to be found.
Output
If there is no solution, output a single integer $0$.
Otherwise, output $r$ rows. The $i$-th of them should contain $c$ space-separated integers, the $j$-th of which is $a_{i,j}$ — the positive integer in the $i$-th row and $j$-th column of a diverse matrix minimizing the magnitude.
Furthermore, it must hold that $1 \leq a_{i,j} \leq 10^9$. It can be shown that if a solution exists, there is also a solution with this additional constraint (still having minimum possible magnitude).
Samples
2 2
4 12
2 9
1 1
0
Note
In the first example, the GCDs of rows are $b_1 = 4$ and $b_2 = 1$, and the GCDs of columns are $b_3 = 2$ and $b_4 = 3$. All GCDs are pairwise distinct and the maximum of them is $4$. Since the GCDs have to be distinct and at least $1$, it is clear that there are no diverse matrices of size $2 \times 2$ with magnitude smaller than $4$.
In the second example, no matter what $a_{1,1}$ is, $b_1 = b_2$ will always hold, so there are no diverse matrices.