In Taiwan, many mathematics teachers design board and card games to help students grasp difficult mathematical concepts. Recently, a particular card game has gone viral among elementary and middle school teachers because it effectively helps students understand the concepts of divisors and multiples, while also being highly engaging for both teachers and students. The rules of the game are as follows. The teacher prepares (n) distinct cards, labeled from (1) to (n). The (i)-th card has an integer value (a_i) written on it, and the integers (a_1, a_2, \dots, a_n) are in a strictly increasing order. There are (m) students labeled from (1) to (m) participating in the game. Before the game begins, each student receives a nonempty subset of the (n) cards. No two students share any card, and at least one card remains undealt. Let (k) denote the number of undealt cards initially. The game consists of (k) rounds. In each round, the following steps occur in order: The teacher selects one of the remaining undealt cards uniformly at random and reveals it to all students. Let (c) be the integer written on this card. Each student simultaneously chooses exactly one card from their own collection. The ownership of the revealed card is determined as follows: Among the values of all cards chosen by the students, consider those that are divisible by (c). If there are one or more such values, the student who selected a card with the smallest divisible value wins the revealed card and adds it to their collection. If no chosen cards have value divisible by (c), the revealed card is discarded (remains unowned). Discarded cards are not used in subsequent rounds. Among the values of all cards chosen by the students, consider those that are divisible by (c). If there are one or more such values, the student who selected a card with the smallest divisible value wins the revealed card and adds it to their collection. If