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Interactive Audio Lesson
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Introduction to Pigeonhole Principle
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Hello everyone! Today we are going to explore a fascinating concept known as the pigeonhole principle. Can anyone tell me what they think this principle implies?
Is it about arranging things into boxes or groups?
Exactly! The pigeonhole principle states that if we have more items than containers, at least one container must hold more than one item. This principle will be crucial in our proof today regarding multiples of integers with only 0s and 1s.
But how will we apply that in our proof?
Good question! We will create a set of numbers and analyze the remainders upon division by an integer n.
Defining the Numbers
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Let’s define some numbers: our first number is 1, then 11, 111, and so on. Who can guess how many of these we will define?
Are we going to define n+1 numbers?
Exactly! This is crucial because we will later apply the pigeonhole principle to this set. How many possible remainders can we have when dividing these by n?
There are n possible remainders!
Right! Since we have n+1 numbers, we know by the pigeonhole principle that at least two of these numbers will yield the same remainder. Let’s see why that’s important.
Analyzing Remainders
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So, if two of our numbers share the same remainder, say x_i and x_j, what can we say about their difference?
Their difference will likely involve zeros and ones?
Exactly! The subtraction x_i - x_j will give us a number comprised of trailing zeros and leading ones. This is where we prove the existence of a multiple of n with only digits 0 and 1.
So that means it will always be divisible by n?
Precisely! Hence, this proof solidifies our statement that for any integer n, there exists a corresponding multiple made entirely of the digits 0 and 1.
Summary and Conclusion
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To wrap things up, we used the pigeonhole principle to find that for any integer n, there is a multiple composed only of 0s and 1s. Can anyone summarize how we reached this conclusion?
We defined a sequence of numbers made only of the digit 1 and compared their remainders when divided by n.
And we learned that at least two numbers will share the same remainder resulting in a number with only 0s and 1s in its formation!
Fantastic! Remember, the next time you face a problem involving distributions, consider using the pigeonhole principle.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section explores the idea that for any integer n, one can find a multiple of n that contains only the digits 0 and 1. Through the application of the pigeonhole principle, the proof establishes that there are multiple combinations of the digit '1' formed into a number, which ensures that the remainder of these combinations when divided by n will yield duplicates, leading directly to a number composed only of 0s and 1s that is divisible by n.
Detailed
Detailed Summary
In this section, we are tasked with proving that for any integer n, a multiple of n exists with a decimal representation consisting only of the digits 0 and 1. To illustrate this, we define a sequence of integers wherein each integer is composed solely of the digit '1', for instance, the numbers 1, 11, 111, etc. By utilizing the pigeonhole principle, we aim to show that among these integers, at least two will yield the same remainder when divided by n.
We start by defining our numbers: 1 (one '1'), 11 (two '1's), and so forth up to a number consisting of (n+1) '1's. The remainders from dividing these integers by n are limited to values between 0 and n-1, thus presenting n possible remainders.
Given that we have (n + 1) numbers and only n possible remainders, the pigeonhole principle guarantees that at least two of these integers will share the same remainder when divided by n. Denote these integers as x_i and x_j. The difference (x_i - x_j) will equate to a decimal number with trailing zeros and leading ones, indicating it is indeed made up solely of the digits '0' and '1'. Since both integers produced the same remainder, it follows that (x_i - x_j) is divisible by n.
Therefore, this confirms the claim that for any integer n, there exists a multiple of n consisting only of the digits 0 and 1.
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Audio Book
Dive deep into the subject with an immersive audiobook experience.
Introduction to Universally Quantified Statement
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So question 10 we want to prove a universally quantified statement. Namely, we want to prove that you take any integer f, there is always a multiple of f which has only the digits 0’s and 1’s in its decimal expansion.
Detailed Explanation
The statement we are trying to prove is that for any integer 'f', we can find a number which is a multiple of 'f' and consists only of the digits '0' and '1' in its decimal format. A universally quantified statement means it must hold true for all possible values of 'f', not just specific cases.
Examples & Analogies
Think of it like finding a special type of lock (the multiple of f) that can be opened using a specific key (the number made up of 0's and 1's). No matter which lock you have (different integers), there exists a key (a corresponding number) that will fit.
Examples for Specific Integers
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Before going into the proof if you want to take few examples say f = 1 then I always have the number 1 which is a multiple of 1 and which has only the digit 1 in its decimal expansion. Remember it is not mandatory that you have both 0’s as well as 1 in the decimal expansion.
Detailed Explanation
For the integer 'f'=1, the number '1' itself is the simplest example of a multiple of 'f', displaying only the digit '1'. If 'f'=2, then '10' is a valid multiple because it is equal to '2 times 5' and uses only 0's and 1's. The case with 'f'=3 and the number '111' shows how we can generate combinations that still adhere to the rule.
Examples & Analogies
It’s like having a vending machine that only gives a specific type of snack (the multiple of f). If you put in different amounts (the integers), you might get different snacks that still follow these rules. For instance, if you always put in 1 dollar, you always get a snack without any extra ingredients (just 1).
The Need for Proof Beyond Examples
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But this is a universally quantified statement and we cannot prove a universally quantified statement just showing examples for a few cases. So we have to give the proof for arbitrary f.
Detailed Explanation
While examples help illustrate the point, they do not provide a formal proof. To establish that the statement is true for any integer 'f', a general proof method is necessary. Thus, we utilize the pigeonhole principle, a fundamental concept in combinatorics that allows us to reason about the existence of certain relationships between a set of items.
Examples & Analogies
Imagine you have a box (the set of integers) and you want to distribute a fixed number of balls (0's and 1's) into it. To say a few boxes already have balls doesn't mean all boxes do. To confirm that every box can have one ball requires you to show a systematic way to do it, just as we do in our proof.
Defining the Decimal Numbers
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Again, we are going to apply here the pigeonhole principle. So let me define a few decimal numbers here. I define the first decimal number to be 1. I define the second decimal number as 11, the i-th decimal number as a decimal number consisting of n number of 1’s and the f+1 decimal number which has the digit 1, f+1 number of times.
Detailed Explanation
In constructing our proof, we create a series of decimal numbers, each consisting entirely of the digit '1'. The first number is simply '1', the second '11', the third '111', and so on. This forms a basis for the possible multiples we will be examining, structured as a function that relates the number of '1's to the integer 'f'.
Examples & Analogies
Think of these decimal numbers like building blocks: each block is a '1'. You can stack 1 block, then 2 blocks, and so on. Just as you can create taller and taller structures, we can create larger numbers, ensuring they still adhere to our foundational rule about digits.
Application of the Pigeonhole Principle
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Now what can I say about this remainders? It is easy to see that these remainders belong to the set 0 to f−1 because of the simple fact that you divide any number by n, the only possible remainders could be 0 if it is completely divisible by f or the remainders could be 1, 2 … f−1.
Detailed Explanation
Using the pigeonhole principle, we recognize that there are 'f' possible remainders when dividing by 'f', but we have 'f+1' numbers created from our decimal representations. This guarantees that at least two of those numbers will share the same remainder when divided by 'f'. This is crucial because the same remainders indicate that we can form a number divisible by 'f'.
Examples & Analogies
Imagine you have a classroom of students (the numbers) sharing a limited number of lockers (the remainders from 0 to f-1). If there are more students than lockers, at least two must share a locker. This overlapping ensures that certain conditions are met, as in our case where two formed numbers yield the same remainder.
Conclusion of the Proof
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So if I take n – m then this will be a decimal number which will have trailing 0’s and then at the leading positions you will have the 1’s. That means it is a decimal number which has only the characters 1’s and 0’s.
Detailed Explanation
The difference between our two numbers that give the same remainder produces a number consisting of only '1's followed by '0's. This structure clearly adheres to the requirement of having only digits '0' or '1' in its decimal form. Furthermore, since the two original numbers were both multiples of 'f', their difference must also be a multiple of 'f'. Hence, we've proven the original statement.
Examples & Analogies
Think of creating a pattern in a painting. If your paint colors are limited to just black (1) and white (0), you can create complex patterns (the numbers) that stick to these colors. No matter how many patterns you create (the multiples), as long as they’re composed of black and white, they will always fit your artistic restrictions.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Pigeonhole Principle: A principle used to show that there must be at least one duplicate in a finite set with limited options.
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Remainders: The outcome when one number is divided by another.
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Multiples: Numbers that result from multiplying an integer by another integer.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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For n=2, the number 10 (which is 1 multiplied by 10) consists of digits 0 and 1.
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For n=3, the number 111 is divisible by 3 and contains only the digit 1.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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Pigeonhole principle, oh so slick, more items than boxes - that’s the trick!
📖 Fascinating Stories
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Once in a village with many chickens (items) but only a few coops (containers), it was found that some chickens shared the same coop. This story illustrates the concept of the pigeonhole principle perfectly.
🧠 Other Memory Gems
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Remember PRM - Pigeonhole, Remainders, and Multiples when solving problems involving distribution.
🎯 Super Acronyms
P-R-M
- Pigeonhole
- Remainders
- Multiples - the key concepts of this section!
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Pigeonhole Principle
Definition:
A principle stating that if n items are put into m containers with n > m, then at least one container must contain more than one item.
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Term: Remainder
Definition:
The amount left over after division of one number by another.
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Term: Multiple
Definition:
A number that can be expressed as a product of another number and an integer.