17.7.1. Universally Quantified Statement

We want to prove that for any integer k, there is always a multiple of k which has only the digits 0’s and 1’s in its decimal expansion.

This statement asserts that no matter which integer you choose (let's call it k), we can always find at least one multiple of k that is composed entirely of the digits 0 and 1 when written out in decimal form. For example, if k = 2, the number 10 is a multiple of 2 and is made up of the digits 1 and 0.

Imagine you have a friend who loves collecting coins. For every integer k they pick, you assure them that they can always find a special coin that represents their favorite number and consists only of a shiny '1' or a dull '0'. This gives them confidence in their coin collection strategy, knowing that no matter how they choose their number, they will always find that special coin.

For k = 1, the number 1 is a multiple of 1 and has only the digit 1 in its decimal expansion. For k = 2, we can take the number 10, which is a multiple of 2 and has only 1’s and 0’s.

Here, the text selects specific values for k (1 and 2) and demonstrates the concept by providing straightforward examples. When k = 1, the multiple is 1 itself, made entirely of the digit '1'. For k = 2, the multiple 10 illustrates how we can use these digits. This easy-to-understand method is not enough to prove the statement universally; yet it efficiently demonstrates the idea for these specific cases.

Think of a vending machine that only dispenses coins with either 1 or 0 written on them. If your friend puts in 1 coin, they get a '1' back (the multiple of 1), but if they put in 2 coins, they get a '10' back, illustrating that with different inputs, they always receive coins made of 1’s and 0’s.

To prove this universally quantified statement, we are going to apply the pigeonhole principle.

The pigeonhole principle states that if you have more items than containers and you want to distribute the items among the containers, at least one container must hold more than one item. In this case, if we define several numbers made up of 1’s and the remainders when divided by k, we can apply this principle to ensure that at least two of them will yield the same remainder, ultimately leading to a number composed of 0’s and 1’s that is a multiple of k.

Imagine you are putting different types of colored balls into a series of boxes. If you have more balls than boxes, you can't avoid placing at least two balls in the same box. Think of the balls as our constructed numbers made of 1's. No matter how you arrange them, eventually, their characteristics will overlap, allowing them to fit the remainder criteria when multiplied and subdivided accordingly.

The resulting number from the difference of two numbers yielding the same remainder forms a decimal number which only contains the digits 0’s and 1’s.

When we take two numbers that yield the same remainder and construct a new number using their difference, this new number can be shown to consist solely of 0’s and 1’s. This is because any number that results from subtracting one constructed number of 1’s from another will have a defining structure with trailing zeros and leading 1’s. This ensures we get non-negative integers that fit our original query of being multiples of k.

Consider a scenario where you're building a tower of blocks. If you have two stacks of blocks, and they both have the same height (this represents our same remainder), when you build a third stack with the difference of their heights, it will only contain blocks with designs made up of 0's and 1's. Thus, regardless of how you stack, you can't avoid creating such structures!

This method confirms that we can always construct a valid number that is divisible by k and consists solely of the digits 0 and 1.

The conclusion ties together the earlier statements, confirming that for any integer k you choose, it is always possible to construct a number out of 0’s and 1’s that remains a multiple of that integer. The proof utilizes the pigeonhole principle and arithmetic properties of remainders to demonstrate the existence of such a number universally.

Imagine you're weaving a cloth that can only use two colors—black and white chips. Regardless of the size or pattern you choose, as long as you keep adding blocks to create your cloth, you'll always be able to complete a design, showcasing the fact that just like weaving materials, numbers can always be structured in a way that satisfies the conditions of the problem.

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