3. Each square on an $8 \times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of 3 , the number of checkers in each column is a multiple of 5 .
Assuming the top left corner of the board is shown below, how many checkers are used in total?
Solution. Answer: 30 checkers. The total number of checkers $T$, must be a multiple of 3 (adding all rows). It must also be a multiple of 5 (adding all columns). 3 and 5 are both prime, hence the only numbers which are multiples of both must be multiples of 15 , hence $T=15 k$ for some $k$. The depicted corner of the board has 2 checkers, hence the total number of checkers is $>0$. Every column has a multiple of 5 , but also has a maximum of 8 , hence the maximum possible number is 5 per column, hence $T \leq 5 \times 8=40$. Hence, $T=15$ or $T=30$. Currently we have two checkers in different columns. Each of these columns must have 5 checkers. Assuming we have 15 checkers in total, this indicates that we have only one remaining non-zero column but both the first and second row need to have two additional checkers in them to reach a multiple of three, leading to (at least) four non-zero columns, ie $T \geq 4 \times 5=20$. The only multiple of 15 in the appropriate range is 30 , therefore 30 checkers have been used.