A13155. Diverse Matrix

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:

$\begin{pmatrix} 2 & 9 & 7\\ 4 & 144 & 84 \end{pmatrix}$ We construct the array $b$ :

1. $b_1$ is the greatest common divisor of $2$ , $9$ , and $7$ , that is $1$ ;
2. $b_2$ is the greatest common divisor of $4$ , $144$ , and $84$ , that is $4$ ;
3. $b_3$ is the greatest common divisor of $2$ and $4$ , that is $2$ ;
4. $b_4$ is the greatest common divisor of $9$ and $144$ , that is $9$ ;
5. $b_5$ 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).

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.