I really liked this video. You gave a good background explanation on a topic which some might find unfamiliar and proceeded to solve the problem. I already know a bit about modular arithmetic but if I didn’t I still feel that I would be able to understand it.
I love all of your videos. It must be really tiring and difficult to write the AMC questions down and erase them again on the whiteboard. You must have dedicated lots of time for all of your videos. Thank you so much! You are helping lots and lots of students.
Hi. Thank you.🙏 It was a bit tiring to do that, but it's also a Passion Project, and when you feel passionate about what you are doing, it feels less like work sometimes. Thank you for the consideration in this thought process. I did see the other comment, and was planning to respond when I could, but I have been very busy, and happened to see this now on a break, so am replying now. Good luck with all of your studies also. 🙂
True which is why being in practice with all of the concepts will increase the likelihood of them to be recognized and resolved more quickly. But under pressure we all sometimes miss things. Shortcuts etc.
Thank you so much Victoria, I think I got a combination of both my parents sense of humor, and I do try to incorporate it in explanations. Thank you for noticing. Happy New Year! 🙂🎉
@@TheBeautyofMath Happy new year too. I am done doing the problems from 2010 to 2020, so now I am doing the problems before 2010, and I found a question (2009 10B p21) that can be solved really fast using your method! ps: I am preparing for amc 10a and b this year!
@@TheBeautyofMath Hi, hope you still remember me haha. I actually did not reload my vpn for a long time so I lost access to youtube. China stuff... yeah I did qualify for the aime, but it was a close cutoff. I will try to improve my math skills this summer to get a better score for the aime next year. :)
@@victorialiao4695 Of course I remember you. Frequent and kind commenters get remembered! I did not however see this comment til just now when I was responding to a recent comment!
a) thanks for uploading, the solution on the AOPS website really sucked and this made it clear! b) so when we do 2^3= 2(mod 6) we notice that the number when modded out is just itself without the cube. What about number like 6^3 or 7^4. Bcuz those numbers are not equal to themselves in mod 6. I think the sequence repeats, but that makes me think that the a1+a2 thing is not equal to a^3 equation. Sorry if this doesn't make sense. If you don't get it I'll try and rexplain my question c) lighting seems fine in this video! It's definitely lighter than the other room and it gives a nice happy vibe to the vid cuz of all the natural light
A) no problem B) 6^3 is 216=0(mod 6) and 6 itself is 6= 0(mod 6). So both 6 and 6^3 are in fact congruent mod 6. It works for all cubed numbers. The 7^4 is a different animal. It happens to be equal to it's modded out value of 1^4 in mod 6. But in general it won't always be equal to itself. What I mean is 7^4= 2401=1(mod 6) and 7= 1(mod 6). But other examples fail like 2^4=16=4(mod 6). While 2= 2(mod 6). So this literally only works (a number is congruent to it's 3rd power) for mod 6 and for the power of 3. What works for all cases is that if I first mod out a number to it's remainder. That remainder to whatever power the first one was to will create the same remainder as the original number to it's original power. Example: 8=3(mod 5) 8^4=4096=1(mod 5) And if I took the original modded out value of 3 and took it to the 4th power. 3^4=81=1(mod 5) but it isn't equal to 3(mod 5) anymore. That property is unique to mod 6 with cubed original values.
For C) yeah the natural light is good, but I usually film at night. So I will need artificial light for that time as previous videos reveal, I was able to make it appear a little brighter in this room by adjusting white balance.
If a1+a2+...+a2018=2018^2018, then we have to reduce 2018^2018 mod 6. 2018=2(mod 6), 2018^2=4(mod 6), 2018^3=2(mod 6), 2018^4=4(mod 6). There is a pattern between 2,4(mod 6) as for every n≥2, if n-odd, it will be 2(mod 6), and n-even it will be 4(mod 6), and hence 2018^2018=4(mod 6). So answer choice E.
This is good and true, but you are not solving the final question. The final question has each a1, a2, a3 being cubed. But you are not in the above explanation showing that the cubed portion has no impact on the final answer. But the method of demonstrating that the 2018^2018 being 4(mod 6) is nice for sure.
@@TheBeautyofMath Oh OK. I saw your solution and it seems very nice! When I did the problem, I guessed it would be 4(mod 6) for a1,a2,a3,...a2018 all cubed. It did save a lot of time when doing the problem though, but I understand your point. In competitions such as USAMO/USAJMO or IMO, the solution would be worth nothing and I would need to explain more.
Not sure if you were being sarcastic or not...but going to assume you meant it literally. If such is the case, I am glad to have offered inspiration. 🙂
Not really. That would have been a more efficient way of saying it perhaps, but referring to something as 11(mod6) simply means we are talking about the remainder when I divide 11 by 6. It is acceptable to use it in this manner, although you are correct to recognize not common. Was my purpose to demonstrate modding out, then certainly the way you state it is the only way it gets stated. So if anything I over-explained perhaps or chose a less time efficient route to express the concept, but it is nonetheless an acceptable way of making the statement.
Weird I replied to this from RUclips and the reply shows on RUclips but not in my RUclips Studio. I am not sure if that means it doesn't notify you or what is going on with that, but basically you can learn about it here: ruclips.net/video/oXqhZBhGAcs/видео.html
Lol my wife hates it 🤣 "no fashion...look horrible!". I am not allowed to wear it with her or she says she will be embarrassed. Lol. I really like it though, always been a fan of the mythical dragons.
@@TheBeautyofMath When I and my 12-year-old son watched this video tonight, as soon as I said "That's a cool T-shirt", my son shook his head and said "What a nerd." I guess we all look at things differently.
Listened. Heard it. Yeah this white board is slightly less quality than the old one, made of different writing surface. I will ask if they can get me a white board I had like in the old room. I apologize for it for the time being. Not much I can do except not film until it's replaced. 😞. A high quality white board of that size can get pretty expensive. Let me do some research on the cost and which ones are easiest to erase. I will get it replaced as soon as possible.
You look so passionate going over this question. I love your enthusiasm!
Thank you! 😊
I really liked this video. You gave a good background explanation on a topic which some might find unfamiliar and proceeded to solve the problem. I already know a bit about modular arithmetic but if I didn’t I still feel that I would be able to understand it.
Thank you for the feedback Neil, it is very helpful!
I love all of your videos. It must be really tiring and difficult to write the AMC questions down and erase them again on the whiteboard. You must have dedicated lots of time for all of your videos. Thank you so much! You are helping lots and lots of students.
Hi. Thank you.🙏 It was a bit tiring to do that, but it's also a Passion Project, and when you feel passionate about what you are doing, it feels less like work sometimes. Thank you for the consideration in this thought process. I did see the other comment, and was planning to respond when I could, but I have been very busy, and happened to see this now on a break, so am replying now. Good luck with all of your studies also. 🙂
@@TheBeautyofMathPassion is wild
I have a passion for math
Wow, this solution is easy to understand, thanks for uploading. On the actual test we only have about 3 min to figure this out from scratch lol.
True which is why being in practice with all of the concepts will increase the likelihood of them to be recognized and resolved more quickly. But under pressure we all sometimes miss things. Shortcuts etc.
@@TheBeautyofMath ,that is so true
you teach like a math comedian!! really like this video!
Thank you so much Victoria, I think I got a combination of both my parents sense of humor, and I do try to incorporate it in explanations. Thank you for noticing. Happy New Year! 🙂🎉
@@TheBeautyofMath Happy new year too. I am done doing the problems from 2010 to 2020, so now I am doing the problems before 2010, and I found a question (2009 10B p21) that can be solved really fast using your method! ps: I am preparing for amc 10a and b this year!
Please leave a comment if you end up qualifying for AIME. I love to hear such things. :)
@@TheBeautyofMath Hi, hope you still remember me haha. I actually did not reload my vpn for a long time so I lost access to youtube. China stuff... yeah I did qualify for the aime, but it was a close cutoff. I will try to improve my math skills this summer to get a better score for the aime next year. :)
@@victorialiao4695 Of course I remember you. Frequent and kind commenters get remembered! I did not however see this comment til just now when I was responding to a recent comment!
I really liked the crash course on modular arithmetic. Thank you!
Very welcome, glad it helped!
Wow, I remember being completely floored by this problem in-contest - great solution!
Thanks Richard!
a) thanks for uploading, the solution on the AOPS website really sucked and this made it clear!
b) so when we do 2^3= 2(mod 6) we notice that the number when modded out is just itself without the cube. What about number like 6^3 or 7^4. Bcuz those numbers are not equal to themselves in mod 6. I think the sequence repeats, but that makes me think that the a1+a2 thing is not equal to a^3 equation. Sorry if this doesn't make sense. If you don't get it I'll try and rexplain my question
c) lighting seems fine in this video! It's definitely lighter than the other room and it gives a nice happy vibe to the vid cuz of all the natural light
A) no problem
B) 6^3 is 216=0(mod 6) and 6 itself is 6= 0(mod 6). So both 6 and 6^3 are in fact congruent mod 6. It works for all cubed numbers. The 7^4 is a different animal. It happens to be equal to it's modded out value of 1^4 in mod 6. But in general it won't always be equal to itself. What I mean is 7^4= 2401=1(mod 6) and 7= 1(mod 6).
But other examples fail like 2^4=16=4(mod 6). While 2= 2(mod 6). So this literally only works (a number is congruent to it's 3rd power) for mod 6 and for the power of 3.
What works for all cases is that if I first mod out a number to it's remainder. That remainder to whatever power the first one was to will create the same remainder as the original number to it's original power.
Example: 8=3(mod 5) 8^4=4096=1(mod 5)
And if I took the original modded out value of 3 and took it to the 4th power. 3^4=81=1(mod 5) but it isn't equal to 3(mod 5) anymore. That property is unique to mod 6 with cubed original values.
@@TheBeautyofMath thanks for the answer
I'm sorry I meant 7^3 not 7^4
For C) yeah the natural light is good, but I usually film at night. So I will need artificial light for that time as previous videos reveal, I was able to make it appear a little brighter in this room by adjusting white balance.
@@TheBeautyofMath ok cool. Idk what all that means but this vid looked pretty good
Np. So everything is clear now? Meaning the understanding is clear(not the lighting)?
Great video!! This really helped me, especially as a 7th grader.
Awesome to hear, thankyou for sharing the feedback!
wow this video is amazing, I really liked the modular arithmetic tidbits. Thank you, TheBeautyofMath
Good to hear Ishaan!
Thanks for this video, learned a lot
infinite respect my bro keep it up
Thank you Daniil! Good luck if you are taking this years tests!
awesome video
Thanks!
If a1+a2+...+a2018=2018^2018, then we have to reduce 2018^2018 mod 6. 2018=2(mod 6), 2018^2=4(mod 6), 2018^3=2(mod 6), 2018^4=4(mod 6). There is a pattern between 2,4(mod 6) as for every n≥2, if n-odd, it will be 2(mod 6), and n-even it will be 4(mod 6), and hence 2018^2018=4(mod 6). So answer choice E.
This is good and true, but you are not solving the final question. The final question has each a1, a2, a3 being cubed. But you are not in the above explanation showing that the cubed portion has no impact on the final answer. But the method of demonstrating that the 2018^2018 being 4(mod 6) is nice for sure.
@@TheBeautyofMath Oh OK. I saw your solution and it seems very nice! When I did the problem, I guessed it would be 4(mod 6) for a1,a2,a3,...a2018 all cubed. It did save a lot of time when doing the problem though, but I understand your point. In competitions such as USAMO/USAJMO or IMO, the solution would be worth nothing and I would need to explain more.
13:31 to 14:27 is so inspirational :)
Glad to know it spoke to you. Have fun on tomorrow's test!
👍 taking it soon 🤞
14.06: Just Inspiring!
Not sure if you were being sarcastic or not...but going to assume you meant it literally. If such is the case, I am glad to have offered inspiration. 🙂
At 5:18 did you mean 11 is congruent to 5(mod 6), not 11(mod 6) is congruent to 5(mod 6)?
Not really. That would have been a more efficient way of saying it perhaps, but referring to something as 11(mod6) simply means we are talking about the remainder when I divide 11 by 6. It is acceptable to use it in this manner, although you are correct to recognize not common. Was my purpose to demonstrate modding out, then certainly the way you state it is the only way it gets stated. So if anything I over-explained perhaps or chose a less time efficient route to express the concept, but it is nonetheless an acceptable way of making the statement.
Very nice
One year later he has new room
Uh what is deserted 🏝 principle?
Weird I replied to this from RUclips and the reply shows on RUclips but not in my RUclips Studio. I am not sure if that means it doesn't notify you or what is going on with that, but basically you can learn about it here: ruclips.net/video/oXqhZBhGAcs/видео.html
Cool looking T-shirt!
Lol my wife hates it 🤣 "no fashion...look horrible!". I am not allowed to wear it with her or she says she will be embarrassed. Lol. I really like it though, always been a fan of the mythical dragons.
@@TheBeautyofMath When I and my 12-year-old son watched this video tonight, as soon as I said "That's a cool T-shirt", my son shook his head and said "What a nerd." I guess we all look at things differently.
Lol. My wife said that too "nerd". She would agree with your son 😆
@The beauty of math, this is slightly invasive but what is your name?😅
Not invasive. James
Dont like the sound the marker was making ...
Interesting. I never noticed. It is a different white board. Let me listen carefully for it. Any particular time stamp you heard it at?
Listened. Heard it. Yeah this white board is slightly less quality than the old one, made of different writing surface. I will ask if they can get me a white board I had like in the old room. I apologize for it for the time being. Not much I can do except not film until it's replaced. 😞. A high quality white board of that size can get pretty expensive. Let me do some research on the cost and which ones are easiest to erase. I will get it replaced as soon as possible.