Not Every Case Gives a Solution Here's the Full Analysis

Added: 2026-05-16

[00:00:00]Okay, so last one. GCD of uh Y and 22 equals 11. I guess the the last last one would be the GCD of Y and 22 is 22, but that's not interesting because anything to the 22 is one. Okay, but notice that this immediately We're going to do this slightly differently. This means [music] that Y is equal to 11. Well, yeah, and we might as well take Y mod 22, right?

[00:00:25]So, it's just between 0 and 21. Then, the only thing whose GCD is 11, but now that tells us that X [music] is equal to 5 to the 11, which is congruent to 5 mod 23. You can check that. You kind of maybe know that because 5 is not a primitive root mod 23.

[00:00:47]>> [music] >> Wait, 5 to the 11? That shouldn't be 5.

[00:00:49]22. I'm sorry, this is 22.

[00:00:52]This is 22.

[00:00:55]Actually, it can be equal to 5 because then the order would all be off. If it's equal to 5, then the order would be divisor of 10, but then Look, it's a primitive root, so we know that the order is 22. See, it's like a whole It's a whole mess, right? Something Something goes wrong with the order and all that kind of Okay, so anyway, it's 22. And now what we need to do is check that 22 to the 22 is congruent to 22. But, what's 22 to the 22 mod 23? You get this?

[00:01:25]This is actually easy to calculate.

[00:01:30]Yeah, cuz it's it's one by Fermat's Little Theorem.

[00:01:39]But, that's incongruent to 22 mod 23. [music] It's -1 to an even power.

[00:01:46]That means this is not a solution.