Modulo Challenge

Prove that

$50^n + 48 \times 99^{3n - 1}$

is divisible by 7 for all positive integers $n$ .

#NumberTheory #Sharky

Easy Math Editor

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Comments

Modulo challenge is child's play by Induction....

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Aditya Raut - 6 years, 10 months ago

Yes, I have it relatively low levelled. I want people to use modulo. I already know there is a solution using induction.

Sharky Kesa - 6 years, 10 months ago

I know you know but what i wanted to know is whether you know what the person in image knows about what you know ,know it ?

Aditya Raut - 6 years, 10 months ago

@Aditya Raut – know-ception.

Sharky Kesa - 6 years, 10 months ago

@Sharky Kesa – $50^n+48\times 99^{3n-1}\equiv 1^n+-1\times 1^{3n-1}\equiv 0\pmod{7}$

Daniel Liu - 6 years, 10 months ago

@Daniel Liu – The solution I was looking for.

Sharky Kesa - 6 years, 10 months ago

@Daniel Liu – I thought the same!!

Kartik Sharma - 6 years, 10 months ago

We know that (a+b)(mod m)=a(mod m)+b(mod m) and (ab)(mod m)=a(mod m) x b(mod m) after knowing this it is a child's play. 50=1(mod 7) that implies 50^n=1(mod 7) now keep that aside 48=-1(mod 7) and 99=1(mod 7) that implies 99^(3n-1)=1(mod 7) that implies [99^(3n-1) x 48]=-1(mod 7) Adding both these equations we get 50^n +[48 x 99^(3n -1)]=0(mod 7) done done done dunna done. P.S:THE LAST REMARK WILL ONLY BE UNDERSTOOD BY INDIANS.

Adarsh Kumar - 6 years, 10 months ago

I can understand it , but ten again, I am Indian. Me re ko patha ha hindi likin me re pas nahi ha hindi keyboard. That's me writing hindi, hope you can understand.

Sharky Kesa - 6 years, 10 months ago

wow in never knew that u were INDIAN .PRETTY SURPRISING!!

Adarsh Kumar - 6 years, 10 months ago

@Adarsh Kumar – Sharky Kesa is a pseudonym. Sort of like Robert Saunders - Benjamin Franklin.

Sharky Kesa - 6 years, 10 months ago

Very good solution..I too had the same one..."The last comment was not really guess-able"...ELucidate?

Krishna Ar - 6 years, 10 months ago

Done dunna done.... is actually " डन डना डन " which is synonym for a ecstatic way of saying "it's done"

Aditya Raut - 6 years, 10 months ago

@Aditya Raut – Fortunately, I can read that. Do you have a Hindi keyboard?

Sharky Kesa - 6 years, 10 months ago

@Sharky Kesa – I used the google translate, that is to input hindi there we have handwritten or even phonetic keyboard. btw i have a software too, it's called "BarahaPad" ,almost all indian languages can be typed by it..

Aditya Raut - 6 years, 10 months ago

@Aditya Raut – OK. Did you understand my Hindi I typed up there?

Sharky Kesa - 6 years, 10 months ago

@Sharky Kesa – indi ?

Aditya Raut - 6 years, 10 months ago

@Aditya Raut – Hindi. Sorry.

Sharky Kesa - 6 years, 10 months ago

@Sharky Kesa – Oh my goddd !!!!! @Sharky Kesa in how many things are you so much expertise attainer !!!

Aditya Raut - 6 years, 10 months ago

@Aditya Raut – In how many things are you so much what? I don't get what you mean by expertise attainer. I live in a very English background. If some sentence doesn't make sense to me, there's probably something wrong. BTW, attainer isn't a word.

Sharky Kesa - 6 years, 10 months ago

@Sharky Kesa – Haha we had that named section in our PACE booklets, it's the exercise which is the toughest in the whole book :P By saying to you i meant simply "expert"

Aditya Raut - 6 years, 10 months ago

@Aditya Raut – I am an expert mainly in math and science (though not really). Calvin Lin is an expert at math. I am nowhere near him. I am alright in Hindi, but exceptional in English. I am above average in sport. Above national average in endurance. But yeah, I am a all-rounder you could say.

Sharky Kesa - 6 years, 10 months ago

@Sharky Kesa – I am a volleyballer :P What sport u play ?

Aditya Raut - 6 years, 10 months ago

@Aditya Raut – Cricket. :D

Sharky Kesa - 6 years, 10 months ago

@Sharky Kesa – I think these talk can better be done on google or email..... may i have your email ?

Aditya Raut - 6 years, 10 months ago

@Aditya Raut – sharkesa@gmail.com, what's yours?

Sharky Kesa - 6 years, 10 months ago

@Aditya Raut – Wow! It did ring a bell but yeah..we Southies aren't much used to it :p @Adarsh Kumar -You a telugu?

Krishna Ar - 6 years, 10 months ago

@Krishna Ar – no i am not a telugu.I am a northie.

Adarsh Kumar - 6 years, 10 months ago

@Adarsh Kumar – OH..Well :)..You study in one of the Narayana /Chaiatanya schools?

Krishna Ar - 6 years, 10 months ago

@Krishna Ar – nopesy i study in a government school.

Adarsh Kumar - 6 years, 10 months ago

@Adarsh Kumar – Seriously???/Then how you so intelligent?

Krishna Ar - 6 years, 10 months ago

@Krishna Ar – in maths particularly number theory i learned it all by myself mainly from brilliant.but other math topics were taught to me by my father.in science i am 0.

Adarsh Kumar - 6 years, 10 months ago

yeah sorry i posted it in a haste.

Adarsh Kumar - 6 years, 10 months ago

We will show that $50^n+48\times99^{3n-1}\equiv 0\pmod {7}$

We have $50\equiv 1\pmod {7}$ , $48\equiv -1\pmod {7}$ , and $99\equiv 1\pmod {7}$

Then, $50^n+48\times99^{3n-1}\equiv 1^n+(-1)\times1^{3n-1}\equiv 0\pmod {7}$

Mas Mus - 6 years, 4 months ago