I understand how the inverse part can be a bit tricky for people to understand, since he did not explain it very clearly, so I'll show you how you can think instead. When you come to the part of 5x congruent to 2 for mod 9, imagine the list of numbers that 2 could be instead of 9. By adding 9 to 2 repeatedly, you'll get said list. The list you get is 2, 11, 20, 29, 38.... Here you see that 20 is divisible by 5, such that we can get that 5x/5 is congruent to 20/5 for mod 9, which is equal to x congruent to 4 for mod 9, and there you have your answer.
@@DavisMaths I am not sure, but this is the method that I found to be the easiest to understand. It's not that difficult if you just write out the list. I did not understand congruence until the idea of lists was mentioned in some youtube video I saw. You can think of it like this, if you have a congruent to b for mod n, the list will be {b, b+n, b+2n, b+3n, ....} This method is useful since it can also be used for squares. For example, if x^2 is congruent to 56 for mod 8, we know that you can not take the square root of 56. But if we write out the list: {56, 64, 72, 80, ...}, we find almost immediately that 64 is a square, as such we can take the square root on both sides and find the answer for x. Hopefully this explanation suffices. Goodluck!
I’m a retired life long learner who has always been interested in maths and number theory/math Olympiad problems is my current buzz. These videos are really great for bringing out the big points that might get missed with self study.
what do we do when the number 56 is raised in a very high power ? For example I was given this one: 8x ≡ 11^41 (mod 51) but I can't figure out how to solve it.
There are. But after n=27, i.e. x=247, the next one is x=256 and that is the same as 4 (mod 252) and we go round the same set of 28 numbers differing by 9 each time. Remember we are solving 149x ≡ 56 (mod 252), so any of the infinite number of solutions will always reduce to the same set (mod 252). It is trivially true that all linear congruences (mod m) which have a solution x will have an infinite number of additional solutions that are congruent to x (mod m), so we conventionally only bother to record the solutions x that lie in the range 0
Inverse (mod 9) has an analogy to rational numbers e.g. (1/2)*2=1 so if you multiply a number by its inverse you get 1. The same situation with (modulo 9). 5*2=1 because 5*2=10 and 10 gives remainder 1 by dividing by 9. That's why 5^(-1)=2
Almost every video I've seen so far has had a different explanation... so far I think this one makes the most sense for me, thank you!
ikrrrr
Same here!!!
Hey bro where are you from
I understand how the inverse part can be a bit tricky for people to understand, since he did not explain it very clearly, so I'll show you how you can think instead. When you come to the part of 5x congruent to 2 for mod 9, imagine the list of numbers that 2 could be instead of 9. By adding 9 to 2 repeatedly, you'll get said list. The list you get is 2, 11, 20, 29, 38.... Here you see that 20 is divisible by 5, such that we can get that 5x/5 is congruent to 20/5 for mod 9, which is equal to x congruent to 4 for mod 9, and there you have your answer.
Is there a way that we don’t need the trial and error method?
@@DavisMaths I am not sure, but this is the method that I found to be the easiest to understand. It's not that difficult if you just write out the list. I did not understand congruence until the idea of lists was mentioned in some youtube video I saw. You can think of it like this, if you have a congruent to b for mod n, the list will be {b, b+n, b+2n, b+3n, ....}
This method is useful since it can also be used for squares. For example, if x^2 is congruent to 56 for mod 8, we know that you can not take the square root of 56. But if we write out the list: {56, 64, 72, 80, ...}, we find almost immediately that 64 is a square, as such we can take the square root on both sides and find the answer for x.
Hopefully this explanation suffices. Goodluck!
Understood
This is also my strategy for solving such congruences. :) I thought I am the only one who makes use of listing the elements/numbers in b(mod n)
I’m a retired life long learner who has always been interested in maths and number theory/math Olympiad problems is my current buzz. These videos are really great for bringing out the big points that might get missed with self study.
Yes I found the same, it can make all the difference. :)
bro looks jacked.. i think bro's worth listening to
😂
140x = 56(°252 )
simplificando
5x = 2(°9 )
5x = 20(°9 )
x = 4(°9 )
Professor Penn, Thank you for a brilliant example on Linear Congruence.
For sure the best explanation out there!
u gotta be capping, his explanation was trash
BRILLIANT! Thank you for helping me see it
The inverse part was a bit rushed, but this video is one of the best on congruences so far. Thanks a lot!
what does the inverse bit mean pls?
Michael Penn, clutchhhhhhh
Much much-needed information is covered. Thank you so much.
Hahahaha.
Why didn't he give the link to the previous video he mentioned?
what do we do when the number 56 is raised in a very high power ? For example I was given this one: 8x ≡ 11^41 (mod 51) but I can't figure out how to solve it.
There won't be more modular arithmetic videos?
Best explanation so far
I think this video was...modul-awesome! Thanks again for sharing.
Nice explanation
I don't really get why there are only 28 solutions, I mean, if you say x=4+9*53 for example it still works
You are right. There are infinite many solutions but only 28 unique solutions mod 252.
Ttooo good both the maths and the back ❤❤
So much good information, unfortunately my brain blew up, I'll try this video again after I master the fundementals more.
Two ways of finding inverse
1. Eulers theorem
2. Extended Euclid's algorithm
Extended Euclid's algorithm is more efficient
Man, just try some numbers first.
Thanks man!
Why are there not infinite solutions of the form x=4+9n?
There are. But after n=27, i.e. x=247, the next one is x=256 and that is the same as 4 (mod 252) and we go round the same set of 28 numbers differing by 9 each time. Remember we are solving 149x ≡ 56 (mod 252), so any of the infinite number of solutions will always reduce to the same set (mod 252).
It is trivially true that all linear congruences (mod m) which have a solution x will have an infinite number of additional solutions that are congruent to x (mod m), so we conventionally only bother to record the solutions x that lie in the range 0
TF zuckerberg explained us this easily
like the way u teach 🤘🤘🤘🤘
dont get the inverse part man
Inverse (mod 9) has an analogy to rational numbers e.g. (1/2)*2=1 so if you multiply a number by its inverse you get 1. The same situation with (modulo 9). 5*2=1 because 5*2=10 and 10 gives remainder 1 by dividing by 9. That's why 5^(-1)=2
@@uzytkownik9616 you are a friend, friend.
Lou ferrigno giving me a math lesson
Perfect example.
4:44 was not a good place to stop. Yet.
Thanks 👍👍 u r genius
Note step not understanding
This question can be done more easily
This video is awesome.
When he says the solutions are separated by 9, what the hell does 9 mean like 9 what?
4+9=13+9=22+9=31,etc
This is really helpful.
helped me a lot!
I mean 5x = 2 (mod 9) 5x = 20 (mod 9) = x = 4(mod 9)
0:05
Excelente,gracias
good.
good
Good
That _would_ have been a good place to stop 😁.
I believe this video is older then when he sticked with that phrase
love it
this guy is the worst explainer i have ever seen...jesus christ
Dfa
There are totally 28 incongruent solutions