Modular arithmetic made easy

Solutions to 8 typical questions. More videos / @randellheyman

[@devilgamer1998]

For those who don't know what he did for question 7, think about a clock that has 9 numbers (for mod 9). If you were to go forward 8 from the starting point (9/0) then it is the same as going backwards one from the same point, or -1. Therefore 8 and -1 are interchangeable in mod 9.

[@RandellHeyman]

Josh Rae Thanks.

[@shehab5781]

@@RandellHeyman does that mean i can always choose to use a value such as -1 instead of the more obvious value like 8 whenever i would like to rewrite it or simplify or for example in example 8 or a similar question could i have instead looked for a value x that would get the value -1 when mod 2 and mod 3? In this case it wouldn't work but in another case?

[@ericzuckerburg6292]

I still don't get it

[@quadriolashubomi6208]

I have get my right it looks so simple and easy to me and i can solve any arithmetic that comes to me and u guys may check how logarithms is simple like arithmetic too

[@grover-]

I thought the answer for 7 was also very vague. Especially when he said its obviously 1. Felt real stoopid at that point. 🤪

[@aswinthomas1318]

Thanks a tonne.

[@deshickpyla3869]

Thank you

[@THEHORSELOVER235]

If I have (mod6) or 4 do I still divide?

[@RandellHeyman]

Which question number?

[@StormWolf01]

I like the video, but you casually transformed an 8 into a minus 1, without any explanation, in question 7.

[@RandellHeyman]

Thanks for the feedback.

[@RandellHeyman]

Just answering another comment and realised that you might find it useful to look at my video modular exponentiation made easy kzbin.info/www/bejne/qoXYiKCZntB4ZrM

[@zariars]

For number 6: Could you say that, 6 and 9 both have the common divisor 3, therefore 6x would be equivalent to 3y (mod 9), so it couldn't be equivalent to 1 (mod 9)?

[@RandellHeyman]

+zariars NIcely thought out. You could do what you suggest. I would express it slightly differently. If a solution exists then 6x=9y+1 for some x and y. Since 6 and 9 are both divisible by 3 we have 3v=3w+1 for some v and w. So 3(v-w)=1 which is impossible. Therefore no solution exists.

[@zariars]

Randell Heyman Nice explanation. Still not that good at explaining things myself... yet =)

[@abhinav6476]

at 5:35 how can u replace 8^100 with (-1)^100?

[@RandellHeyman]

Well 8 is equivalent to -1 modulo 9 since the remainder when we divide 8 by 9 is the same as when we divide -1 by 9. Alternatively 8 is equivalent to -1 modulo 9 because the difference between 8 and -1 is 9 which is a multiple of 9. Next, you can multiply modular equations. That is, if 8 ≡ -1 (mod 9) then 8r ≡ (-1)r (mod 9). This in turn means that you can raise both sides of a modular equation to a power. That is, if 8 ≡ -1 (mod 9) then 8^r ≡ (-1)^r (mod 9). That is what I have used. Hope that helps.

[@ablebodied175]

For number 5, can -2 be another answer. I am getting it through Euclids extended algorithim. Basically get 13 = 6(2) + 1. Apply extended algorithim here you get 13 -6(2) = 1. This satisfies Bezouts Theoreom. So I hit the equation with mod 13 and get 0 - 12 = 1. -12 mod 13 = 1. So can -2 be an answer?

[@RandellHeyman]

Yes. -2 is an answer. In fact your method is well suited to harder problems with larger numbers. I didn't use your method here as it is an introduction to modular arithmetic. Just make sure you read questions carefully. Sometimes inverse questions ask for the smallest positive answer to the problem.

[@ablebodied175]

@@RandellHeyman Wow thanks for responding after so many years. I did some research and realised that -2 is congruent to 11 so they basically represent the same number in mod 13. Thanks again!

[@RandellHeyman]

If you are interested, I have a video, modular inverse made easy, that explains 'your' method.

[@abigailakoto6648]

Please explain to us how you got all the answers. I don't understand

[@RandellHeyman]

For question 1 the backet (mod 3) means that numbers are replaced by their remainder when dividing by 3. So it is perfectly correct to say 4 is equivalent to 1 modulo 3 since 4 and 1 have the same remainder when divided by 3. Similarly you could say 1234 is equivalent to 1 modulo 3 since 1234 and 1 have the same remainder when divided by 3. So the answer is 1 since 1 is the smallest positive integer that is equivalent to 1234 modulo 3. In question 2 the idea of looking at remainders works for negative numbers as well. Observe that -27=-6x5+3. So -27 has a remainder of 3 when divided by 5. We could say that -27 is equivalent to 3 modulo 5. So our answer is 3. Once you get the first 2 clear in your mind you should be able to progress through the other questions. Hope that helps.

[@UnderMiles]

jesus this helped me a lot. I just hate how books on algebra just jump any explanation about modular arithmetic and start proving really important stuff by using it.

[@RandellHeyman]

I remember thinking the same when I was studying.

[@rikenm]

For number one, it is x=1 or x ≡ 1, which is x ≡ 1(mod 3)?

[@RandellHeyman]

I think the correct answer is x=1 as it asks for the smallest positive integer solution.

[@spheleleemmanuel7046]

37 mod 5 = 2 how was this done

[@lauge5]

Beacause 5*7=35 and then you have a remainder of 2 because 37-35=2 it could also be 5*8=40 37-40=-3 but 2 is a better number to work with, you use that

[@d12kiem7]

3:16 why is modulur 9 equal to 0?

[@RandellHeyman]

When we are working modulo 9 any number that is divisible by 9 can be replaced by 0. So that's what I do. I replace 999999 and 99999 with 0.

[@danielmcgrane8608]

Modular arithmetic made hard

[@RandellHeyman]

Thanks for the feedback. If you tell me where you are stuck I can try to help further.

[@quadriolashubomi6208]

Is good but u are skipping questions but I still use my brain to take it

[@stefanmarjanovic4768]

yes, you are skipping content without explanation

[@RandellHeyman]

Stefan Marjanovic Thanks for the feedback.

[@samirnoori767]

very unclear solutions. Just goes direkt to the answer.

[@RandellHeyman]

Sorry you didn't understand it. Probably best to find another video, but if you let me know where you first get stuck I'm happy to explain further.

[@faridabdi3590]

The explanation for question one does not make sense to someone who knows nothing about this subject, the syntax and operations performed are inexplicable: Because Randell says 5 mod 3 = 2 mod 3 Without any explanation of why that is or how such a stupid looking syntax came about. Therefore not worth even trying to look at the other questions he aims to explain

[@RandellHeyman]

Sorry the video didn't help, and thanks for letting me know. If you can't find help elsewhere email me through my website and I'll try to help further.