I've never heard of the extended euclidean algorithm before, but this seems super interesting so I'll learn it :) Here is how I solved it, btw (before watching the video): 123x + 45y = gcd(123, 45) Simplify that to 41x + 15y = 1 let's say k = -y (I just wanted to think of it this way, it's not necessary tho) so 41x - 15k = 1 (41x - 15k) - 3(15x) + 3(15x) = 1 -4x - 15k + 45x = 1 4x + 15k - 45x = -1 let's say k = 3x + n (This is because we want that x term to cancel and n is just some number) substitute that into the previous equation: 4x + 15(3x + n) - 45x = -1 4x + 15n = -1 now it's much easier so after a little guess and check, we can see 44 - 45 = -1 works. (It's only one of many other examples) that means x = 11 and n = -3 because earlier we said that k = 3x + n and k = -y so y = -3x - n y = -3(11) - (-3) = -33 + 3 = -30 We see that (x, y) = (11, -30) works, so yay. Thanks for the awesome vid!
00:12 Which "previous video"? This video here is one of the oldest videos on your channel (the 38th video), and the last one about number theory was the 24th video "The Euclidean Algorithm Example 2". I don't see any such proof in that video :q The 37th video is on an entirely different subject (exact differential equations), and there are no other Number Theory videos between 24 and 38.
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It only took me 3 minutes to understand this one. My teacher explained it and I didn't get anything from what she has taught. Thanks man✊
I've never heard of the extended euclidean algorithm before, but this seems super interesting so I'll learn it :)
Here is how I solved it, btw (before watching the video):
123x + 45y = gcd(123, 45)
Simplify that to 41x + 15y = 1
let's say k = -y (I just wanted to think of it this way, it's not necessary tho)
so 41x - 15k = 1
(41x - 15k) - 3(15x) + 3(15x) = 1
-4x - 15k + 45x = 1
4x + 15k - 45x = -1
let's say k = 3x + n (This is because we want that x term to cancel and n is just some number)
substitute that into the previous equation:
4x + 15(3x + n) - 45x = -1
4x + 15n = -1
now it's much easier so after a little guess and check, we can see 44 - 45 = -1 works. (It's only one of many other examples)
that means x = 11 and n = -3
because earlier we said that k = 3x + n and k = -y so y = -3x - n
y = -3(11) - (-3) = -33 + 3 = -30
We see that (x, y) = (11, -30) works, so yay.
Thanks for the awesome vid!
Nice method
Been struggling to understand this for a while and you made it beautifully clear, thank you man
bro this is literally amazing !
really helped me out
man this was absolutaly beautiful
youre an angel sent from heaven! thank you so much, I went from not getting anything to understanding completely! :)
Professor M. Penn. thank you for an excellent lecture on The Extended Euclidean Algorithm.
00:12 Which "previous video"? This video here is one of the oldest videos on your channel (the 38th video), and the last one about number theory was the 24th video "The Euclidean Algorithm Example 2". I don't see any such proof in that video :q The 37th video is on an entirely different subject (exact differential equations), and there are no other Number Theory videos between 24 and 38.
I also had the same question, but there is this newest video of him proving the same result: ruclips.net/video/E_mU_fCmcxc/видео.html
Is this the Mark Zuc of math?
Thank uuu I finally understand what does Extended Euclidean Algorithm means! Very helpful
Thanks for our knowledge and understanding with another exceptional video!
very good explanation
Thanks a lot. Really helped
Thank you! I ask you to make a video about gcd properties, please.
Thank you very much....
I think there are infinitely many solution (solved using modular arithmetic)
But only one solution has x,y in the set of integers
@@parthanaved3866 nah
@@parthanaved3866 try: x=41, y=-112
You can to x add 45n then from y remove 123n. So x = -4 +45n and y = 11 -123n where n is any integer
I prefer to start with gcd and then go "upwards" in the Euclid algoritm
But that is NOT A UNIQUE SOLUTION, IS IT?
Correct. There are infinitely many solutions of the form x=-15n-4, y=41n+11, for n in Z. The solution in the video corresponds to n=0.