# Help Me Out With These 2 Tough Questions

• evanx
In summary, the conversation discusses two mathematical problems and the use of the Well-Ordering Principle in solving them. The first problem involves finding natural numbers that satisfy a certain equation, while the second problem deals with the existence of students with the same number of friends. The conversation also provides hints and guidance on how to approach these problems.
evanx

Q1
m, n are natural numbers (which include zero for the sake of this question) and n =/= 0, then there are natural numbers q and r such that m= qn + r and r < n. Use the Well-Ordering Principle to prove this fact.

Q2
There are 51 students in a class. Prove there are at least two students with exactly the same number of friends.

Last edited:
the well orderin principle says there is a smallest number of a certain type. So in this problem look for the desired number which is supposed to be small. take all numbers like it and among them take the smallest one and see if it is small enough to work for you.

problem 2 is false without assuming each person has at least one friend.

what have you done so far? the first one is a pretty straightforward application of the wop. you've got a set of possible r's which is a subset of N... do i need to say the rest? then you need to show that the smallest element is the "right size", and that yer r & q are unique (uniqueness should be pretty routine, even for a beginner - it's done like every other uniqueness proof)

hmm... I think problem 2 is just fine without any added assumption. Well you have to assume friendship is mutual. If a is a friend of b then b is a friend of a. And no one is their own friend. But these are reasonable assumptions. So my two big hints are: 1)there is nothing special about 51, any number will do, and 2) the pigeon hole principle. So try the same question with 3,4,5 students.

Hope that helps,
Steven

fourier jr said:
what have you done so far? the first one is a pretty straightforward application of the wop. you've got a set of possible r's which is a subset of N... do i need to say the rest? then you need to show that the smallest element is the "right size", and that yer r & q are unique (uniqueness should be pretty routine, even for a beginner - it's done like every other uniqueness proof)

I have got Q2 (sort of...) but am at a loss for Q1 since I am unfamiliar with the Well-ordering principle. I do not have to prove uniqueness of r and q though (I think).

## 1. What are the two tough questions?

The two tough questions refer to a specific set of questions that the person needs help with. It could be related to any topic or subject matter.

## 2. Can you provide the answers to the tough questions?

As a scientist, my area of expertise may not necessarily cover the specific topic of the tough questions. However, I can assist in guiding the person towards finding the answers through research and critical thinking.

## 3. How do I approach difficult questions in general?

The best approach to difficult questions is to break them down into smaller, more manageable parts. This will help in understanding the question better and finding relevant information to answer it.

## 4. Is there a specific method or strategy to finding answers to tough questions?

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## 5. Can you recommend any resources for finding answers to difficult questions?

There are many resources available, depending on the subject matter of the tough questions. Some general resources include libraries, online databases, and scholarly articles. It is also helpful to consult with peers and professors for additional guidance.

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