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?

Rationale

When the player divides the marbles into 5 piles, she will have 3 marbles left over.

The player has a number of marbles that, when divided by 2 and 3, results in no remainders, indicating that the total number of marbles is a common multiple of 2 and 3. However, when dividing by 4, there is a remainder of 2, which gives us clues about the total number of marbles. After calculating the possibilities, we find that when divided by 5, 3 marbles remain.

A (0) YOUR ANSWER

If the player had 0 marbles left over when divided by 5, that would imply the total number of marbles is a multiple of 5. However, given the conditions established with the remainders from divisions by 2, 3, and 4, this cannot be the case, as the total number does not align with being divisible by 5 without a remainder.

B (1) YOUR ANSWER

Having 1 marble left over when divided by 5 would indicate that the total number of marbles is one more than a multiple of 5. However, our calculations based on the given conditions show that the number of marbles results in a remainder of 3 when divided by 5, disqualifying this option.

C (3) CORRECT

This choice is correct as the total number of marbles, which is between 10 and 20, meets the criteria that result in a remainder of 3 when divided by 5. This can be verified by checking the possible numbers (12, 15, 18) that fit all earlier conditions and confirming the division by 5.

D (2) YOUR ANSWER

If there were 2 marbles left over when divided by 5, this would imply that the total number of marbles is two more than a multiple of 5. However, based on the established conditions, the total number aligns with having a remainder of 3, making this option invalid.

Conclusion

The player has a specific number of marbles that fit the criteria of being divisible by 2 and 3 without remainder and leaving a remainder of 2 when divided by 4. Upon further examination, it becomes clear that when divided by 5, the remainder is 3, confirming that the total number of marbles appropriately meets all stated conditions.

When the player divides the marbles into 5 piles, she will have 3 marbles left over.

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