Concepts Covered: Combinatorial Construction and Counting

Hello, young mathematicians! Today, we’re going on a magical journey with a number, 2023. We’re going to explore a fun problem that will make you think and use your math skills in a creative way. Ready? Let’s dive in!

Imagine you have a chalkboard, and you write a positive integer on it. Now, you can perform a magic trick on this number. If the number on the chalkboard is the sum of two positive integers, you can erase it and replace it with the product of these two integers. For instance, if you have the number 7 (because 7 = 2 + 5), you can erase it and write 10 (since 10 = 2 * 5).

Now, the question is, which positive integers can you start with so that after a few magic tricks, you end up with the number 2023?

Let’s break this down.

Firstly, any number greater than 1, let’s call it $a$, can be broken down as $a = (a-1) + 1$. According to our magic trick, this can be replaced with $(a-1) * 1 = a - 1$. So, any number greater than 2023 can be transformed into 2023 by performing this trick multiple times.

Secondly, any number greater than or equal to 2, let’s call it $b$, can be broken down as $b = (b-2) + 2$. This can be replaced with $(b-2) * 2$. Since $(b-2) * 2 = (b-2) + (b-2)$, if $b > 4$, then $(b-2) > 2$. So, $(b-2) * 2 > b$. This means that any number greater than 4 can be transformed into a larger number using our magic trick. Once a number reaches or exceeds 2023, it can be transformed into 2023 using the first trick we discussed.

Let’s illustrate this with an example. Let’s start with the number 5:

$5 -> 2+3 -> 23 -> 6 -> 3 +3 ->33->9->4+5->45->20->10+10->1010->100->30->29+71->29*71->2059->2058->…->2023$

So, any positive integer greater than 4 can be transformed into 2023 after a finite number of steps using our magic trick. Isn’t that amazing?

That’s it for today’s mathematical journey, young wizards! Keep practicing your magic (math) tricks and have fun exploring the magical world of numbers!