Zoe has won some $10 cards as a third prize in a competition she entered. She has decided to give them away to her friends. If she shares them among 3 friends there will be 2 left over, if she shares them among 5 friends there will be 1 left over, if she she ares them among 2 friends there will be none left over. What is the smallest number of phone cards she could have won?
To find the smallest number of phone cards Zoe could have won, we need to find the least common multiple (LCM) of the numbers 3, 5, and 2, and then add 1 to the result.
The LCM of 3 and 5 is 15, because it is the smallest number divisible by both 3 and 5. However, we also need to consider the condition that there are 2 cards left over when shared among 3 friends.
The next multiple of 15 after 2 is 17. Therefore, Zoe must have won at least 17 cards in order for there to be 2 left over when shared among 3 friends.
Similarly, we need to consider the condition that there is 1 card left over when shared among 5 friends. The next multiple of 15 after 1 is 16. Therefore, Zoe must have won at least 16 cards in order for there to be 1 left over when shared among 5 friends.
Finally, we need to consider the condition that there are no cards left over when shared among 2 friends. The common multiple of 17 and 16 is 272. Therefore, Zoe must have won at least 272 cards in order for there to be none left over when shared among 2 friends.
Thus, the smallest number of phone cards Zoe could have won is 272.
To find the smallest number of phone cards Zoe could have won, we need to look for a number that satisfies the given conditions. Let's start by considering the three scenarios:
1) If Zoe shares the cards among 3 friends, there will be 2 left over.
2) If Zoe shares the cards among 5 friends, there will be 1 left over.
3) If Zoe shares the cards among 2 friends, there will be none left over.
Let's use a systematic approach to find the smallest number that meets these conditions. We'll start with the first scenario:
1) If Zoe shares the cards among 3 friends, there will be 2 left over.
This means the number of cards must be 2 more than a multiple of 3. We can set up an equation:
Number of cards = 3n + 2, where n is a positive integer.
Next, let's consider the second scenario:
2) If Zoe shares the cards among 5 friends, there will be 1 left over.
This means the number of cards must be 1 more than a multiple of 5. We can set up another equation:
Number of cards = 5m + 1, where m is a positive integer.
Now, let's consider the third scenario:
3) If Zoe shares the cards among 2 friends, there will be none left over.
This means the number of cards must be a multiple of 2. We can set up a final equation:
Number of cards = 2k, where k is a positive integer.
To find the smallest number of phone cards that satisfy all three scenarios, we need to find the smallest common solution to these equations. This solution can be found by finding the least common multiple (LCM) of the given numbers (3, 5, and 2 in this case).
The LCM of 3, 5, and 2 is 30. This means the smallest number of phone cards Zoe could have won is 30.
To verify this, we can substitute this number into the equations:
1) 30 = 3 * 10 + 2 (satisfying the first scenario)
2) 30 = 5 * 6 + 1 (satisfying the second scenario)
3) 30 = 2 * 15 (satisfying the third scenario)
So, Zoe could have won a minimum of 30 phone cards.
geez, 11 minutes?
impatient much?