Скачать или смотреть Factorials Unfolded: Crafting a Perfect Square out of the Product of the First Hundred Factorials

Hey there, math enthusiasts, Happy Thanksgiving!

In this video, I dive into a fascinating mathematical challenge: find the lone factorial that, when removed, transforms the product of the first 100 factorials into a perfect square. It's a puzzle that might sound daunting at first, but stick with me, and you'll see the beauty in its complexity.

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The approach I take begins not by wrestling with gigantic numbers, but by playing a clever game with exponents when we write the product out differently. We can write that giant factorial product as 1^100 * 2^99 * 3^98 and so on, all the way to 100^1. Here's where the fun begins: I notice that all the even powers (the perfect squares) can be set aside. They're already contributing nicely to our quest for a perfect square.

The real action lies in the odd powers, all of which are conveniently on even bases. For example, 2^99 has just one '2' too many to be a perfect square. Similarly, 4^97 has just one '4' too many. This pattern repeats all the way through, turning our problem into a quest to deal with these pesky even bases.

Since it's all the even numbers up to 100 that are causing those problems, we want to investigate different ways to represent the product of those even numbers. There is actually a related function called the double factorial that represents this: 100!! And that double factorial can then transformed into 2^50 times 50!. Since 2^50 is already a perfect square, our culprit in disturbing the perfect square peace is none other than 50!.

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