Modular Arithmetic (Part 1)

[@anmol3457]

Thanks for these videos. My discrete prof teaches like he's explaining the concepts to himself rather than first time learners, these videos really help.

[@rajupujari6773]

yes, both are valid, first -> -8 = 7(mod 5) formula here is (a = b(mod n)) => (a-b)/n => -8-7/5 => remainder is 0 => first one is valid second -> -3 = -8(mod 5) => -3-5 = -8 => second on is valid

[@kuttypaiyanreshanth8722]

Explain without formula a-b/n

[@sakriyapyakurel7664]

Both are valid For first : -8-7 = -15 (divisible by 5) For second: -3-(-8) = 5(divisible by 5)

[@sravyasirikonda1117]

How to find if a modular arithmetic is valid/ invalid? Simple. a=b (mod n) only if (a-b)/n is divisible.

[@winniebhardwaj3852]

tysm, this helped me alot!!!

[@thor1626]

Thanks man :)))

[@Venkata_krishna_tutorials]

@sravya sirikonda Rhank you

[@top_10_2.O]

ohkk nice

[@msktheabbasi4818]

damn, thanks bro

[@mahdi-hasan]

Note: remember, a = b (mod m) is same as b = a (mod m). 1) -8 = 7 (mod 5) => 0 = 15 (mod 5) [Adding 8 on both sides] => true. [15 mod 5 = 0] 2) -3 = -8 (mod 5) => 5 = 0 (mod 5) [Adding 8 on both sides] => true. [5 mod 5 = 0]

[@aniketpatel-cb1fc]

smart

[@fightforfitness2256]

Thanks bro

[@silviapaolaherrerapasos5410]

No entendi tu respuesta, -8 = 7(mod 5) => (a-b)/n => -8-7/5 = -3 no? no es igual a 0

[@Carrymejane]

How do u know this?

[@mahdi-hasan]

@@silviapaolaherrerapasos5410 "a (mod m)" and "a mod m" is not the same. Google to know the difference.

[@akankshasonkar5845]

Thanks for this i tried your concepts for 5 hrs and now i mastered 🙏

[@nayang9155]

I never put comments because this account is from my mom, but OH MY GOD, this video is crazy, my teacher always says things he believed everyone knows but it was my first time to see those modular congruence for AMC 10 competitions for a freshman, Thank you!

[@bhavyas3023]

Both are valid first one using the property n|(a-b) and the second one using the property a cong b(mod n) => b cong a(mod n)

[@davidbarker5411]

Exceptional video. This is how teaching is done. Thank you!

[@darshanshah9623]

5*(-3) + 7 = 8 5*(1)-8 = -3 hence both are Valid Congruent .

[@alyalyalyalyalyalya]

THANK YOU I COULDN’T UNDERSTAND THIS FOR THREE DAYS

[@otomofrancis7534]

10:26 the first is INVALID while the second is VALID

[@jhin5588]

Best channel ever.

[@PrithaMukherjee-c8l]

Sir, Both are valid. First one, -8 divided by 5, if the quotient used is -3 then the remainder is coming 7 so the expression is coming valid. For the second, -3 divided by 5, the quotient is 0 then the remainder is -3 which can also be written as -8 (also be represented as -8 + 5=-3). So the second expression is also coming valid. Regards.

[@rafastyles1002]

Damn I never expected that a clock was the perfect example of modular operations. Like 22 == 10 (mod 12) Thank you for explaining. I had seen this before in training for math olympiads and never understood

[@RitikTUF]

a congruence b (mod n) represents n divides (a-b) like i) 15 cong 3 (mod 12) ==>> (15-3)/12 =1 it is divisible similarly -8 cong 7 (mod 5) ==>> (-8-7)/5 = -15/5 = -3 it is also true similarly -3 - (-8) = 5 / 5 = 1 also true. hence both last questions are valid.

[@bossthekhani]

I really understand valid and invalid congruence 1) -8==7(mode 5) wrong 2) -3==-8(mode 5) right

[@a.studiotech7393]

Superb explanation

[@carlosgarcia1468]

Both are valid. -n mod m = m - ( n mod m) or n|(a-b), a mod m == b is a== b mod m, a/m gets a remainder = rem when b/m.

[@MaheshKumar-vi7pi]

I am still unable to understand it. Can you describe it in more simpler way?

[@moneebss]

@@MaheshKumar-vi7pi a=b (mod n) only if (a-b)/n is divisible, try this method and you will understand it

[@adityaprasadsahoo924]

The congruence -8 ≡ 7 (mod 5) is true, as both -8 and 7 leave the same remainder when divided by 5.

[@AzharKhan-fb6qv]

At 8:52 you said 38 is congruent to 14 as well in modulo 12. Relating this to our clock analogy , that would mean 38 means 14 aclock but there is no 14 aclock! Plz help.

[@SatyamEdits]

dont relate everything to clock....clock was just an application of it ..but by mathematics it is valid....(may be having application somewhere else than the clock)

[@godseguy]

don't relate it with the clock example

[@akshaygamez4129]

a=km+b it is same as euclid's division lemma (a=bq+r)

[@KenderGuy]

I saw a math puzzle using modular arithmetic and couldn't solve it, so I looked up some videos to brush up was skipping through most of the video to see where my knowledge ends, I hope part 2 has new information for me I was checking and got all the answers right that you showed, but you said to put my answer to the last two in the comments so I decided to humor it the last two are both valid because 5 * (-2) + 2 = -8, therefore -8 mod 5 is congruent with 2 7 is also a multiply of 5, plus 2, so it is also congruent mod 5 with 2 same with the last one, because -5 + 2 = -3 and -10 + 2 = -8

[@IronPulse60]

they are both valid using m/a-b is easier to prove

[@creativeartifact]

In congruence you said that 'a' divide by 'm' give remainder 'b' but this is the defination of 'amodb =r' then why we use congruence

[@narendarbommera.narendar6649]

-8 = 7 (mod 5) -3 -8 (mod 5) both are valid

[@AhmadJZaigham]

Greatly helped, thank you!

[@satyamy866]

a cong b(mod m) means a=km+b, so k value are -3 and 1 which is integer so both are valid.

[@joshwzr1257]

..no

[@satyamy866]

@@joshwzr1257 @📞 Why?? Answer:: Both are valid ( -8cong(7 mod5), and -3 cong -8(mod 5) )

[@joshwzr1257]

@@satyamy866 ..is it possible to divide a negative number by positive and we get a remainder?

[@satyamy866]

@@joshwzr1257 Yes, you will always get remainder(it might be +ve -ve Or 0) if u divide a number(that is + or - or 0) by non-zero numbers.

[@joshwzr1257]

@@satyamy866 ..thanks for the reply Sir. so should I now ignore the fact that a remainder can't be negative? Solong as it is a dividable number?, I have just been confused about that

[@vinaybhaikikani7660]

Nice explanation

[@shaylasharmin7202]

Very clear explanation 😃

[@leonardonifinluri4416]

It's valid. Let = be congruence sign. -8 = 7 (mod 5) means when we -8 is divided by 5 the reminder is 7.why? -8 =(equal sign) 7-15 =(equal) -3×5 + 7.

[@kirankharel1288]

im confused.... for -8 congto 7(mod 5), (-8-7)/5 is divisible so it can be said as congruent but since -8/5 is remainder -3 and -8 isnot equal to 7+5 it doesnt seem valid....HOW??

[@vasulibhai5202]

1-) -8%5 --> -8+5 = -3 --> -3+5 = 2 and 7%5 => 2. Hence correct 2-) -3%5 --> -3+5 = 2 and -8%5 --> -8+5 = -3 --> -3+5 = 2. Hence correct.

[@sarazahoor9133]

The first test question is invalid and the second is valid, is it correct?

[@hometours3638]

both are valid

[@rufaidashaik4753]

yes

[@PhenomenalONE1]

both are valid. 5*(-3)=-15 and -8+15 gives 7 as remainder. for the 1st one

[@potatootter5088]

@@PhenomenalONE1 "-8+15 gives 7 as remainder" Where did you get 15? Thanks.

[@thirumal1086]

use this rule : a ≡ (b mod n ) only if n divides (a-b)

[@vishnupalve3982]

Thank you sir❤

[@lekaotamoeta2171]

Thank you so so much!

[@sanas.193]

Thank you so much ❤

[@lvsleav1ng_]

First question for us to answer is invalid, while the next is valid. But i keep seeing comments saying both are valid. May i ask how -8 is congruent to 7 (mod 5) is valid?

[@CHAI_tan_ya]

hey... im wondering the same thing... did you figure out? if yes do tell..

[@gayathrikathiri8027]

@@CHAI_tan_ya Same for me. We need explanation.

[@anmol3457]

consider = to be congruent. Now, x = y (mod n) only when (x - y) is completely divisible by n. Now, -8 = 7 mod 5 is valid because -8 -7 is equal to -15 (which is completely divisible by 5). You can use the same concept in the second example (and every other example). Edit: from the internet - The formal definition of modular congruence is: x≡y(modn) This is equivalent to saying that there exists an integer k such that: x−y=kn In your example, when you substitute x = -8, y = 7, and n = 5, you get: (−8−7)=−15=(−3)×5 So, indeed, -8 ≡ 7 (mod 5) is valid because -15 is divisible by 5.

[@Trithi-x4m]

Sir, if we divide 2 by 5 and the quotient is 1 then is the remainder -3 valid or the quotient has to be 0 and the remainder 2 is just represented as -3??

[@jacquelinesanchez8483]

(1) q = -1 -8 div 5 = -1, -1x5 = -5 -8 mod 5 = -8 -(-5) = -3 q = -2 q= -2, -2x5 = -10 -8 mod 5 = -8-(-10) = 2 q = -3 q = -3, -3x5 = -15 -8 mod 5 = -8-(-15) = 7

[@Carrymejane]

How do u know this?

[@SatyamEdits]

Solution of question :- example 2 mod 5 = -3 or 2 why...??? because when we wrap around we can either travel forward from the multiple of or backward from another multiple of 5 so if we travel forward (from 0) we get answer 2 or if we travel backward (from 5) we get answer as -3 ...both are same. now applying same analogy to last two questions.. -8 mod 5 = 2 or 7 or so on... for 2 we travel backward (from 10) and for 7 we travel backward (from 15) ...so it is valid... 2. -3 mod 5 = -3 or -8 ...

[@creativeartifact]

Actually there explaination is incomplete that why u stuck there at 2== -3(mod5). In congruence never use 2mod5 instead use a-b is divisible by m or not if yes then valid otherwise not.

[@nectar_swaminarayan]

@@creativeartifact thanks it is helpful 👍😀

[@nectar_swaminarayan]

thanks helpful

[@souhardyagayen3040]

Yes Sir, you are right both are VALID Because -8 mod 5 = -3 (addition) or 2 [a+b=(-3)+(5)] or 7 [a+b=2+5] or 12 [a+b=7+5]........ -3 mod 5 = 2 (addition) or -3 [a-b=(2)-(5)] or -8 [a-b=(-3)-(5)] or -13 [a-b=(-8)-(5)]..........

[@dipendersingh9495]

Last 2nd -> INVALID Last -> VALID

[@IT.Diwakar]

@Neso Academy, plz solve................< -8 ≅ 7 (mod 5) > 5*(-3) +7 =15 but the question is , is multiplying by negative integer valid? Also, (-8)-(7) =-15 but is it okay to say -15 is divisible by 5? Even when reduced we get -3 ≅ 2 (mod 5) which doesnt feel comfortable, 2 ≅(-3) (mod 5) this is 100% valid and gives comfort.

[@misupartha3118]

Both two are valid.

[@anjalisrivastava3640]

Answer of last two questions are valid congruence.

[@nononnomonohjghdgdshrsrhsjgd]

yes

[@joshwzr1257]

@@nononnomonohjghdgdshrsrhsjgd ..but why?

[@alyaalsalamah2151]

How come 38=14mod12 is valid !!! when I divide 38/12 = 3.166666 then 3x12 =36 and then 38-36 = 2 not 14 ????!!!! I don't get it please explain

[@kuttypaiyanreshanth8722]

Changing the quotient and remainder

[@kuttypaiyanreshanth8722]

38 /12 = (12*2 times= 24) = 38-24= 14 . Quotient= 2 Remainder=14

[@rahulsbytes]

If two numbers are divided with same number and getting same reminder is called "congruence".

[@laylay-l6y]

both valid

[@souhardyagayen3040]

Both are VALID Because -8 mod 5 = -3 (addition) or 2 [a+b=(-3)+(5)] or 7 [a+b=2+5] or 12 [a+b=7+5]........ -3 mod 5 = 2 (addition) or -3 [a-b=(2)-(5)] or -8 [a-b=(-3)-(5)] or -13 [a-b=(-8)-(5)]..........

[@md.tanvirulabsarchowdhury178]

Both are Valid

[@codejns2275]

07:37 slip of tongue :-2 times 24😅 he is correct but it sounds funny

[@gourabmukhopadhyay7211]

Hello How come first one be valid???if we are dividing -8 by 5 then the reminder is "-13" , not "7"..I am confused.

[@ananyachakraborty4413]

both are valid

[@geralenegarcia5296]

they are both valid

[@sshashank1895]

Both the unsolved examples are valid, right?

[@souhardyagayen3040]

Yes, Both are Valid

[@SatyamEdits]

first invalid 2nd valid

[@ikechukwuamalu878]

The last two examples are very much congruent

[@potatootter5088]

Hi, can somebody show me how --8 congruent 7 (mod 5) is valid? Thank you.

[@Vanbaan]

-8 (mod 5) is the same as -3 (mod 5) which is the same as 2 mod (5). 2 mod(5) has remainder 2 (2/5). 7 mod(5) also has remainder 2 (7/5), thus -8 congruent to 7 mod(5)

[@srishylesh2935]

-8 mod 5 is 2 since 10-8=2

[@reeddhikundu8776]

7 when divided by 5 gives remainder 2 and -8 when divided by 5 gives remainder 2 (5*-2+2=8) . So 8 is congruent to 7(mod 5)

[@anmol3457]

consider = to be congruent. Now, x = y (mod n) only when (x - y) is completely divisible by n. Now, -8 = 7 mod 5 is valid because -8 -7 is equal to -15 (which is completely divisible by 5). You can use the same concept in the second example (and every other example). Edit: from the internet - The formal definition of modular congruence is: x≡y(modn) This is equivalent to saying that there exists an integer k such that: x−y=kn In your example, when you substitute x = -8, y = 7, and n = 5, you get: (−8−7)=−15=(−3)×5 So, indeed, -8 ≡ 7 (mod 5) is valid because -15 is divisible by 5.

[@pzsstudio7822]

There you go

[@SusanthaWijesinghe-xp8rw]

38 isn't congruence to 14(mod 12) as I understand. Pls, explain if i'm wrong.

[@Study-d3x2w]

38/12 remainder = 2 14/12 remainder = 2 Thats is why 38 congruento 14(mod 12)

[@kerry-ch2zi]

Why didn't they teach this to me in third grade? Congruence is so much more useful and real than equality...

[@ABDULRAHMANYAKOOB-b7u]

wait a second 38 = 14(mod12) is invalid , it should be 38 = 2(mod12) isn't it?

[@Carrymejane]

It's multiple by 3 yes 2(mod13) but if u calculate from multiple by 2 it's 14(mod12) pls watch them carefully 😄

[@AfnanAl-Nuzaili]

First invalid Second valid

[@joshwzr1257]

..yes

[@avinashmaurya3625]

Another way to validate mod: a := b mod n => a mod n = b mod n

[@1maruf]

*I presume, Last Two are Valid coz if a mod n = r1, b mod n = r2 and r1==r2 then we can say a ≡ b ( mod n)😎🤓🧐*

[@MSRproduction2066]

thanks

[@rajeshprajapati4863]

Answer to H.W : Both are Valid Congruence.

[@bhimmu-uj6lb]

i used windows+shift+S to snip &copy to word i succeeded first time second time 2nd screen it did not work why ?

[@joshwzr1257]

..second last invalid, last valid. Is it true sir?

[@aradhanamurarka2949]

Both are invalid - 8 mod 5 && -3 mod 5 is 2

[@asikurrahmanrifat1364]

Those 2 ar valid

[@TekkenGamer7]

-8 invalid -3 is valid

[@alokendughosh7013]

Both are valid.

[@pradeepmewada7126]

I couldn't understand last two can anyone give me simple solution. And explainaetion

[@captain_vegan]

use the formula a = km + b

[@atishsinhaa]

Reminder cant be greater than diviser so 2nd one is not valid

[@izilinegbe]

How to now if valid???

[@santhipriyapriya9828]

Thank you...

[@arjunchourasia6524]

for anyone lost , let me explain. so we're given that -8=7(mod5) i have a very easy trick which you can use... step 1 = take the remainder (7) to RHS so it will become -8-7=mod5 which gives us -15=mod5 and let this be (1) step 2 = divide the rhs with the mod no, in this case, divide -15/5, which gives us remainder 0 step 3 = ignore the quotient and cross check whether you really are getting 0 as remainder in (1) here, we can see that there is no value for remainder here, so therefore the given set of modular arithmetic in valid. im still a highschooler in 10th grade india, so if you can point out any mistakes in my method, please do point out. thanks

[@godseguy]

where is part 2?

[@stevelinson593]

Arrigato

[@mdrizwan2931]

1 is valid and 2 is invalid.

[@mdrizwan2931]

Solutions of second if we try to multiply by any positive num. Or negative number a always odd number came and remender is in even negative number. That's not possible to say it's valid

[@dipparekh3594]

there is a mistake 12*3 is 36 so for the equation you have given 38 congruent 14 mod 12 its not possible since 12 can go into 38 3 time leaving a remainder of 2 so how are u getting a remainder 14

[@KoKo-xj9jl]

-2+10 = 8

[@stevelinson593]

Hi 0:22

[@stevelinson593]

Arigatto

[@manimala-yr7pz]

y difficult task giving to viewers

[@Onlinedailyfacts]

both are invalid i think,

[@alt.redwanahmed5310]

Twal!

[@saketh712]

yes no

[@OkaforChinemerem-zq6un]

The 6th answer wasn't correct Should be 2= 1(mod 5) and not 2 because 2/5 have a remender of 1. 2= 6(mod 5) and not -3

[@ToanPham-wr7xe]

😮

[@thornthanlerdhirunwong2212]

falsw and true

[@alirezakiani5863]

💙

[@risegiy5647]

bro skipped first grade 💀thats not how u pronounce hour

[@subharthigayen8371]

Both are VALID Because -8 mod 5 = -3 (addition) or 2 [a+b=(-3)+(5)] or 7 [a+b=2+5] or 12 [a+b=7+5]........ -3 mod 5 = 2 (addition) or -3 [a-b=(2)-(5)] or -8 [a-b=(-3)-(5)] or -13 [a-b=(-8)-(5)]..........

[@abhigayen7046]

Both are VALID Because -8 mod 5 = -3 (addition) or 2 [a+b=(-3)+(5)] or 7 [a+b=2+5] or 12 [a+b=7+5]........ -3 mod 5 = 2 (addition) or -3 [a-b=(2)-(5)] or -8 [a-b=(-3)-(5)] or -13 [a-b=(-8)-(5)]..........

[@BasavarajChougala-n5n]

Both are valid

[@AgrawalTushaarRohitTemasekjc]

both are valid

[@agamanigayen6510]

Both are VALID Because -8 mod 5 = -3 (addition) or 2 [a+b=(-3)+(5)] or 7 [a+b=2+5] or 12 [a+b=7+5]........ -3 mod 5 = 2 (addition) or -3 [a-b=(2)-(5)] or -8 [a-b=(-3)-(5)] or -13 [a-b=(-8)-(5)]..........

[@abhigayen5966]

Both are VALID Because -8 mod 5 = -3 (addition) or 2 [a+b=(-3)+(5)] or 7 [a+b=2+5] or 12 [a+b=7+5]........ -3 mod 5 = 2 (addition) or -3 [a-b=(2)-(5)] or -8 [a-b=(-3)-(5)] or -13 [a-b=(-8)-(5)]..........