Intereting Posts

What distinguishes the Measure Theory and Probability Theory?
For complex matrices, if $\langle Ax,x\rangle=\langle Bx,x\rangle$ for all $x$, then $\langle Ax,y\rangle=\langle Bx,y\rangle$ for all $x$ and $y$?
Using right-hand Riemann sum to evaluate the limit of $ \frac{n}{n^2+1}+ \cdots+\frac{n}{n^2+n^2}$
An entire function is identically zero?
Derivation of the density function of student t-distribution from this big integral.
Characteristics method applied to the PDE $u_x^2 + u_y^2=u$
Volterra integral equation of second type solve using resolvent kernel
Why is $(1+\frac{3}{n})^{-1}=(1-\frac{3}{n}+\frac{9}{n^2}+o(\frac{1}{n^2}))$ and how to get around the Taylor expansion?
Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous
Eigenvalues of product of two hermitian matrices
Angle of reflection off of a circle?
A good introductory discrete mathematics book.
Does $\pi$ have infinitely many prime prefixes?
Why are Lie Groups so “rigid”?
I Need Help Understanding Quantifier Elimination

We have a set A of numbers 1, 2, 3… to 200

The question is asking me to prove that if I choose 101 numbers from the set, there will be two such that one evenly divides the other.

I know this could be the pigeonhole principle question. I could prove by contradiction that no two numbers will evenly divide each other. Assume I take 101 numbers, I can’t take all the odd numbers because there is only 100 of them, so there will be an even number. I think this goes no where.

- Another pigeonhole principle question
- Proof that Fibonacci Sequence modulo m is periodic?
- 101 positive integers placed on a circle
- Pigeonhole Principle Question: Jessica the Combinatorics Student
- Combinatorics Pigeonhole problem
- Pigeonhole Principle to Prove a Hamiltonian Graph

Using a direct proof if I choose 101 numbers, I will get either 100 even + 1 odd or 100 odd + 1 even. ~~In order for two numbers to evenly divide each other I would choose the 100 even, and there is a big probability that two will be even, but if I have 100 odd + 1 even, there will be only 1 even… So I’m not sure how to solve this…~~

- Pigeon hole principle with sum of 5 integers
- How are the pigeonholes calculated in this pigeon-hole problem?
- Polygon and Pigeon Hole Principle Question
- Prove that if two miles are run in 7:59 then one mile MUST be run under 4:00.
- pigeonhole principle and division
- Distribution of points on a rectangle
- regarding Pigeonhole principle
- Pigeonhole principle application
- pigeonhole principle divisibility proof
- The pigeonhole principle - how to solve questions like that?

Pick $101$ elements from $A$, label them $a_1,\ldots, a_{101}$. We can assume that $a_1 < \ldots < a_{100} < a_{101}$. Since we have $101$ distinct elements, $a_1 \leq 99$.

Consider the set of remainders upon division by $a_1$. Since $a_1 \leq 99$, there are at most 99 such remainders. Let $r_2$ be the remainder upon dividing $a_2$ by $a_1$, $r_3$ the remainder upon dividing $a_3$ by $a_1, \ldots, r_{101}$ the remainder upon dividing $a_{101}$ by $a_1$.

We have $100$ remainders $r_1,\ldots ,r_{101}$ (pigeons), and at most $99$ possible remainders (pigeonholes) upon dividing by $a_1$. Thus, by the pigeonhole principle, $r_i = r_j$ for some $2\leq i < j \leq 101$. Now what can you say about the number $a_j – a_i$?

**Hint:** The boxes will have labels $1$, $3$, $5$, and so on up to $199$. Odd labels! Note that there are $100$ boxes.

Box 1: Contains $1,2,4,8,16, 32,\dots$

Box 3: Contains $3,6,12,24, 48, \dots$

Box 5: Contains $5,10,20, 40,\dots$.

And so on.

Box 99: Contains $99,198$

Boxes 101, 103, and so on are pretty boring. Box 101 only contains the number $101$, Box 103 only contains $103$, and so on.

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- Geometry Construction Problems
- Linear independence of fractional powers
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- Solving recurrence relation?
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- Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…)
- What is the sum of the prime numbers up to a prime number $n$?
- What is the word for a corollary that follows from a proof?
- If xy + yz + zx = 1, …
- Fractions in Ancient Egypt
- How to find the complex roots of $y^3-\frac{1}{3}y+\frac{25}{27}$
- Pair of straight lines
- An example of noncommutative division algebra over $Q$ other than quaternion algebras
- Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers