1. **Stating the problem:** We have three decks of cards with 30, 75, and 42 cards respectively. We want to find which of these decks can be evenly distributed among 2, 3, 5, or 10 players. 2. **Formula and rules:** A deck can be evenly distributed among $n$ players if the number of cards is divisible by $n$. This means the remainder when dividing the number of cards by $n$ is zero. Mathematically, for a deck with $d$ cards and $n$ players, $d \mod n = 0$. 3. **Check divisibility for each deck:** - For 30 cards: - Divisible by 2? $30 \div 2 = 15$, remainder 0, so yes. - Divisible by 3? $30 \div 3 = 10$, remainder 0, so yes. - Divisible by 5? $30 \div 5 = 6$, remainder 0, so yes. - Divisible by 10? $30 \div 10 = 3$, remainder 0, so yes. - For 75 cards: - Divisible by 2? $75 \div 2 = 37.5$, remainder not 0, so no. - Divisible by 3? $75 \div 3 = 25$, remainder 0, so yes. - Divisible by 5? $75 \div 5 = 15$, remainder 0, so yes. - Divisible by 10? $75 \div 10 = 7.5$, remainder not 0, so no. - For 42 cards: - Divisible by 2? $42 \div 2 = 21$, remainder 0, so yes. - Divisible by 3? $42 \div 3 = 14$, remainder 0, so yes. - Divisible by 5? $42 \div 5 = 8.4$, remainder not 0, so no. - Divisible by 10? $42 \div 10 = 4.2$, remainder not 0, so no. 4. **Summary of results:** | Deck | 2 players | 3 players | 5 players | 10 players | |-------|-----------|-----------|-----------|------------| | 30 | Yes | Yes | Yes | Yes | | 75 | No | Yes | Yes | No | | 42 | Yes | Yes | No | No | **Final answer:** - 30 cards can be distributed evenly among 2, 3, 5, and 10 players. - 75 cards can be distributed evenly among 3 and 5 players. - 42 cards can be distributed evenly among 2 and 3 players.