(n+1)(2n+1)=6k^2. Let n+1=2p^2 and 2n+1=3q^2.So k=pq.Eliminating n ,we get x^2-3q^2=1 which is a Pell’s equation (Here x= 2p). x=26 and q=15 are the solutions of x^2-3q^2=1 .So n=337 and k=195. Second solution is n=2465845537 .Here p=35113 and q=40545 and k=pq =1423656585.
Bro here you have assumed that 2n + 1 is a multiple of 3. But what if it is not? It is possible that (n + 1) itself contains both 2 and 3 in its prime factorisation
7:51 Bhaiya binomial theroem me remainder wale question karte hai to hum use multiple±1 ke term me break karte hai aur power n raise karte hai yaha n 1 ho gayega aur agar n2 'd' se divide ho rha hai to 12 n2 bhi hoga to use hume multiple ±1 ke form me tod liya aur to remainder 1/d agya to humne prove kar diya dono co prime hai
I solved it using python code for i in range(1, 1000): # start from 1 to avoid division by zero s = (((i * (i + 1) * ((2 * i) + 1)) / 6) / i)**(1/2) if s.is_integer(): print(i, s) # else: # print('no')
To the people demotivating him in the replies... I mean just see, bro just did something very cool. Bro naturally talented in coding and isn't it a good thing that a person has some hobbies other than runnin' in the ratrace?
Yeh sab maine socha tha, like literally co primes hoyenge and all that, lekin aage kaise badhe woh nhi ho paya rha tha, waise without substitutions bhi hojata I guess
14:55 maine tg pr apna solution bheja tha mai bhi usme iss conclusion tk aa gya tha , just after this step we can convert it into pell's equation for a more synthetic solution. Anyway good Question
Bhaiyya, this is similar to a BMO 2 question 1 in the 90s i guess (probably 1992-93). I am very sure that you would have taken this question from an olympiad based resource!. Happy Problem solving :)
bhaiya yeh sawaal wahi solve kar payega basic number theory use karke jo baith ke khelta ho numbers se paper pe, without using any hardcore number theory
My approach (N+1)(2N+1) =6K^2 Therefore (N+1)(2N+1) sould be divisible by 6 Let N = 6a+r where r belons to {-2,-1,0,1,2,3} Satisfying values we get r=1 or r=-1 Therefore N=6a+1 or 6a-1 Taking first case Then (6a+2)(12a+3)=6k^2 (3a+1)(4a+1) = K^2 Ab same process but is equation ko lane me itna dimag lagane ki jarurat nhi h iske bad vhi dono co prime h to dono ko perfect sqaure hona padega simultaneously for which we get a=56 Therefore N=6×56+1 =337.
Honestly idk abt the question but the solution was really baseless, baseless in the sense ki it won't help any drop adding to ur iq or in ur jee it was just a typical Olympiad question with a non so formal solution, for me it was a waste of time, im saying all this because of why u were so happy with ur soln
Indeed, thoda out of the box chalagaya par the fun part is koi typical identity use ni hui. But i don’t agree with IQ vala point and raw creativity it holds is bound to increase ones depth.
@@jeesimplified-subject iq wala point can be added to the one jo ye manipulation hi krta hai sirf jisko aur new cheeze mili but as a jee aspirant it doesn't help right
@@adityarajkanth why do you think so one dimensionally? Do you need to do each and every thing that perfectly "aligns with the syallabus" and "helps" in JEE? Can't there be room for enjoyment of mathematics?
The answer is n = 337, for m = 56.
Also, send us the hardest question you have ever tried here 👇🏻
forms.gle/HZxgUAKdWV1Pywgb8
*n=337
Bhaiya how are you able to say :
n2 is integral
U said n2 is(n1)/3
I have sended a fantastic problem which is of moderate to difficult level but is extremely tricky
Pls make the next video on the problem that I have sended😊❣️
That's exactly the solution i posted in group.... It's still there you'll can check
Mene ise bina pells k hi kia tha i used number theory
Congrats, If you cracked it 😄
@air02IITB haa me whi hu
@air02IITB nhi bhai me sirf 2025 aspirant hu olympiad wagera k bare me kch pta nhi h mujhe filhal btw thank you
Can you tell your solution?
@@anaygoyal1657 it's of one nd half a page, how do i share that to u?
(n+1)(2n+1)=6k^2. Let n+1=2p^2 and 2n+1=3q^2.So k=pq.Eliminating n ,we get x^2-3q^2=1 which is a Pell’s equation (Here x= 2p). x=26 and q=15 are the solutions of x^2-3q^2=1 .So n=337 and k=195. Second solution is n=2465845537 .Here p=35113 and q=40545 and k=pq =1423656585.
Bro here you have assumed that 2n + 1 is a multiple of 3. But what if it is not? It is possible that (n + 1) itself contains both 2 and 3 in its prime factorisation
@@FocusonSolution780 fir 2n+1. 4k-1 ki form mai ayega joki kabhi bhi perfect square nhi hai
7:51 Bhaiya binomial theroem me remainder wale question karte hai to hum use multiple±1 ke term me break karte hai aur power n raise karte hai yaha n 1 ho gayega aur agar n2 'd' se divide ho rha hai to 12 n2 bhi hoga to use hume multiple ±1 ke form me tod liya aur to remainder 1/d agya to humne prove kar diya dono co prime hai
16:50 Isme 196 bhi ek possibility hai for 3m+1
I solved it using python code
for i in range(1, 1000): # start from 1 to avoid division by zero
s = (((i * (i + 1) * ((2 * i) + 1)) / 6) / i)**(1/2)
if s.is_integer():
print(i, s)
# else:
# print('no')
😂
dekho isko 🤦♂🤣
Bhai exam me command prompt kholega💀
To the people demotivating him in the replies...
I mean just see, bro just did something very cool. Bro naturally talented in coding and isn't it a good thing that a person has some hobbies other than runnin' in the ratrace?
Btw keep up bro!
GCD(k,12k-1)=GCD(k,12k-1-11k)=GCD(k,k-1)=1 because GCD of two consecutive numbers is always 1
please don't get disappointed by less views and stop posting please. love you
Stop posting??? You good buddy?
@@pulkitgupta5550 he w anted to say " please dont stop posting because of less views"
17:11 196 is also a possible candidate for 3m+1 right?
we gotta find for n minimum
Yeh sab maine socha tha, like literally co primes hoyenge and all that, lekin aage kaise badhe woh nhi ho paya rha tha, waise without substitutions bhi hojata I guess
And i was thinking ki mujhe hi video nahi mil rahi😂 post hi aaj kari hai😵💫
Am I the only one jisko yeh solution dekhke darr lag raha hai 💀
Solution hai kaha!!!???? Samajh hi nhi ara
Edit : mil gaya
Yes
@@anuragsingh9767 yeh video he solution hai 💀
Ha sirf Tu hi he
Definitely not😂
Raw math indeed
14:55
maine tg pr apna solution bheja tha mai bhi usme iss conclusion tk aa gya tha , just after this step we can convert it into pell's equation for a more synthetic solution. Anyway good Question
end mein values daal ke solve kr dia... proper solution waali feel nahi aai... uske aage bhi thodi number theory laga ke proper solve kr skte the
Bhaiyya, this is similar to a BMO 2 question 1 in the 90s i guess (probably 1992-93). I am very sure that you would have taken this question from an olympiad based resource!. Happy Problem solving :)
not similar this is that one only
8:53 What if n2 is 1?
Interesting question bhai sare gyan chakshu khul gaye
maza baandh diya bro
bhaiya yeh sawaal wahi solve kar payega basic number theory use karke jo baith ke khelta ho numbers se paper pe, without using any hardcore number theory
Indeed, Fair point
I also posted right answer in group
Bhai ye intro problem level h oly k liye
bhaiya maine root me 6 ko 2 x 3 me baant liya aur usse hit and trial se 337 hi ans aaya under 5 min
Bring more such problems
Getting 40-50 marks in maths jee advance 2024, jee mains 98%ile in maths
17 marks in jee mains April attempt
But 40 marks in adv 2024
@@dictetord12 nice
@@dictetord12 Kitni rank aa jayegi bro
16:33 196 bhi hoga 3m+1 form ka😢😢😢😢😢
Aur multiplication me satisfy krega kya voh?
Bro which white board software you are using
Same ques bro
Bhai ni smjh aaya 15:39
Iske liye kon kon se chapters clear hone chiye
I'm in 11th
this is a British Mathematics OLympiad qs
My approach
(N+1)(2N+1) =6K^2
Therefore (N+1)(2N+1) sould be divisible by 6
Let N = 6a+r where r belons to {-2,-1,0,1,2,3}
Satisfying values we get r=1 or r=-1
Therefore N=6a+1 or 6a-1
Taking first case
Then (6a+2)(12a+3)=6k^2
(3a+1)(4a+1) = K^2
Ab same process but is equation ko lane me itna dimag lagane ki jarurat nhi h iske bad vhi dono co prime h to dono ko perfect sqaure hona padega simultaneously for which we get a=56
Therefore N=6×56+1 =337.
196 bhi 3m+1 ke type ka hai
Nice question!!❤
We aren't taught number theory.
Amazing
Maza aagaya kasam se
Pells equation se do minute me solve ho jayega
i am into iitgbjust for no reason i am watching this
Bhai ye to number theory ka question hai
Me khud ek solution kara hoo bo kya share karuun ?
Ha
❤
😵😵😵kuch samajh nahi aya
Honestly idk abt the question but the solution was really baseless, baseless in the sense ki it won't help any drop adding to ur iq or in ur jee it was just a typical Olympiad question with a non so formal solution, for me it was a waste of time, im saying all this because of why u were so happy with ur soln
Indeed, thoda out of the box chalagaya par the fun part is koi typical identity use ni hui.
But i don’t agree with IQ vala point and raw creativity it holds is bound to increase ones depth.
@@jeesimplified-subject iq wala point can be added to the one jo ye manipulation hi krta hai sirf jisko aur new cheeze mili but as a jee aspirant it doesn't help right
@@adityarajkanth why do you think so one dimensionally? Do you need to do each and every thing that perfectly "aligns with the syallabus" and "helps" in JEE? Can't there be room for enjoyment of mathematics?
@@divyanshgarg5434 bhai phir tu kuch zyada hi free h
@@adityarajkanth You can say that, as I had started my prep in 9th
Bhai tujhe samjhana ni ata bilkul smjh ni aya solution dekh k mtt smjhaya kr
Heh... NEET maths level question 🤡
by the way one solution is 65521 also