(Solved): 2. Let \( a \) and \( b \) be positive integers with \( a \) a factor of \( b \). Show that an \( m ...

2. Let \( a \) and \( b \) be positive integers with \( a \) a factor of \( b \). Show that an \( m \)-by- \( n \) board has a perfect cover by \( a \)-by-b pieces if and only if \( a \) is a factor of both \( m \) and \( n \) and \( b \) is a factor of either \( m \) or \( n \). (Hint: Partition the \( a \)-by- \( b \) pieces into \( a \) 1-by-b pieces.)