17. A player has between 10 and 20 marbles on the floor. She divides the marbles evenly into 2 piles, and then divides them evenly into 3 piles. When she tries to divide the marbles evenly into 4 piles, she has 2 marbles left over. If she divides the marbles into 5 piles, how many marbles will be left over, if any?
Answer: C
The player will have 3 marbles left over when dividing into 5 piles.
When the player divides her marbles into 5 piles, she will have 3 marbles left over based on the conditions provided about her total number of marbles.
A) 1
This option is incorrect because if the player had 1 marble left over when dividing into 5 piles, the total number of marbles would not satisfy the conditions of the problem. The total marbles must be congruent to 2 when divided by 4 and cannot yield a remainder of 1 when divided by 5.
B) 2
While this option seems plausible at first, it is incorrect. If the player had 2 marbles left over when dividing by 5, it would contradict the given condition that she has 2 marbles left when dividing by 4. The total number of marbles must align with the established remainders from the previous divisions.
C) 3
This option is correct. Given that the player has between 10 and 20 marbles, and knowing the conditions of division into 2 and 3 piles, we can deduce that when the total is divided by 5, it will indeed leave a remainder of 3. This fits all the conditions outlined in the problem.
D) 2
This option is incorrect as well, as explained in option B. If the player had 2 marbles remaining when dividing into 5 piles, it would not conform to the established conditions regarding the remainders from the divisions into 2 and 3 piles, leading to a contradiction.
Conclusion
The correct answer is clearly option C, which states that 3 marbles will be left over when dividing into 5 piles. This conclusion is supported by the remainders from the previous divisions, confirming that all other options fail to meet the conditions set forth in the problem. Thus, option C is the only valid response.