Given the clues ... solve the crossword

First, we know that $b$ is either $5$ or $0$ (since is divisible by $5$ ). $b$ cannot be $0$ because it would not satisfy the second down statement( $bd$ must be divisible by $13$ , so b must be more than $0$ ), so b is equal to $5$ . If $b=5$ , then $d=2$ ( $52$ is divisible by $13$ ). Then, $c$ must be $4$ ( $42$ is divisible by $7$ ). This time, $a$ can be equal to $2$ or $8$ . The problem states that there is no same digits in the four variables, so $a=8$ . Thus, $a+b+c+d=5+2+4+8=19$ .