There doesn't seem to be enough information to answer problem 1.
I would ask your teacher for clarification.
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Problem 2
Answer: 47 cards
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Step-by-step explanation
x = number of cards, which is some positive whole number.
Your teacher has provided these facts:
- If I deal among four people, three cards remain.
- If I deal among three people, two cards remain.
- If I deal among five people, two cards remain.
which I'll call "fact 1", "fact 2", "fact 3" in that order.
For now I'll focus on fact 3.
Recall that multiples of 5 either end with 0 or 5. Example: 25 is a multiple of 5, and so is 60.
Fact 3 says that we get a remainder 2 when dividing by 5. Therefore, x must be something like 25+2 = 27 or 60+2 = 62. I added 2 to each previous example mentioned earlier.
In short, the value of x must end with 2 or 7.
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Now focus on fact 1.
The multiples of 4 are {4, 8, 12, ...} all of which are even since 2 is a factor of each.
Add remainder 3 to each item to get {7, 11, 15, ...} and it should be fairly obvious that all of these items are odd.
What does this mean? It means we know 100% that x ends with 7 in the units digit. If x has 2 in the units digit, then x would be even, but it's not in the set {7, 11, 15, ...}
So this is very useful info that x ends with 7.
x could be any of these: {7, 17, 27, 37, ...}
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We've looked at facts 1 and 3 so far. The only thing left is fact 2.
List the multiples of 3 to get
{3, 6, 9, 12, 15, 18, ...}
then add on the remainder 2 giving this new updated set
{5, 8, 11, 14, 17, 20, ...}
We see that 17 is in that set which fits the criteria of "the number ends with 7". So it's possible that 17 is the answer.
But 17/4 = 4 remainder 1 which doesn't fit fact 1 (since we want remainder 3 instead).
But we can add on multiples of 3*5 = 15 to extend to this new set of
{17, 32, 47, 62, 77, ...}
then erase stuff that ends with 2 to get
{17, 47, 77, ...}
Through guess and check, 47 will work out as shown in the next section below. Therefore, 47 is the final answer.
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Check:
- 47/4 = 11 remainder 3
- 47/3 = 15 remainder 2
- 47/5 = 9 remainder 2
We get the proper remainders mentioned earlier, so the answer has been confirmed.
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Side note: The convention for problems like this is to use the Chinese Remainder Theorem. However, I decided not to use it because I wasn't sure if your teacher has covered modulo arithmetic yet or not. If s/he has covered it, then I recommend using that method instead of what I posted above. It's a much more efficient option.