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A short class with Sarah and her students. The other ways are under More.
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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.
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.
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.
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.
Pigeonhole principle, oh so slick, more items than boxes - that’s the trick!
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.
Remember PRM - Pigeonhole, Remainders, and Multiples when solving problems involving distribution.
P-R-M: Pigeonhole, Remainders, Multiples - the key concepts of this section!
Term
Pigeonhole Principle
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Term
Remainders
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Term
Multiple
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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.
The amount left over after division of one number by another.
A number that can be expressed as a product of another number and an integer.
In this section, the pigeonhole principle is used to demonstrate the existence of multiples of an integer that consist solely of the digits 0 and 1 in decimal form.
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.
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.