Cute Cubes
Consider positive integers a k where k ∈ ( 1 , 2 , 3 , 4 , 5 ) .Suppose we have that 9 ∣ a 1 3 + a 2 3 + a 3 3 + a 4 3 + a 5 3 .Then we must necessarily have that n ∣ a 1 a 2 a 3 a 4 a 5 irrespective of choice of ( a 1 , a 2 , a 3 , a 4 , a 5 ) .
Determine the least value of n where n > 1 .
The answer is 3.
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1 solution
This solution is flawed.
It is nowhere mentioned in the problem that the 5 cubes must leave different residue classes modulo 9.It is quite possible that all of them may leave the same residue class.Thus one cannot apply PHP here.We could have applied PHP only when we were certain that all 5 cubes leave different residues modulo 9.
It seems the solution proposer needs a better understanding of the PHP.
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See this is obvious that if the sum of cubes is divisible by 9 and only 3 residues are there then one of thef leaves a remainder 0, but for better understanding you see, I have edited the solution. And i think that you shouldn't care about privacy as i very well know who you are and hey KKFL am I right.
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I am sorry...I am not able to recognize you.Have we met ? And who is this ayon and KKFL you talk about ? Sorry if I upset you but I really dont recall having an acquaintance named Shreyansh.
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@The Dark Lord – Hey man are you serious, I mean you follow a guy named Ayon and tell that you don't know him.
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@Shreyansh Mukhopadhyay – Yes I actually don't know him personally just came across one of his problems and liked it very much !! However if you are upset with him then I agree to comply with you and stop following him.
There, its done :) !!
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@The Dark Lord – @Shreyansh Mukhopadhyay And BTW please enlighten me about that KKFL thing ? Is it supposed to be something related to some game ?
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@The Dark Lord – Why do you have so much interest on KKFL and I'm not upset with any named ayon. You don't know how to be good actor and pretend well.
@The Dark Lord I very well know that you are Ayon Ghosh.
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@Shreyansh Mukhopadhyay like I said I am least interested in KKFL but highly interested in SJFL ;) :D ?!
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@The Dark Lord – @The Dark Lord I am unable to understand what are you talking about and yeah KKFL was a typo.
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@Shreyansh Mukhopadhyay – Ummm...Yea like I said we are making no sense.So good night.
In case you are wondering...SJFL was a typo too ! God knows why I wrote it. ;).
@Shreyansh Mukhopadhyay Please tell me Madhu shabdroop quick if u arent sleeping...
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@Shreyansh Mukhopadhyay Bhai plz 12 bajne wale hain aur mere abhi bhi 5 shabdroop bache hain.Kuch to batao kahan se karu ye 5 ?!!! Net me nahin mil rahe hain !!
http://www.hindi2dictionary.com/shabd-roop/madhu.html
We are aware of the fact that a ≡ 0 , 1 , − 1 mod (3)
⇒ a 3 ≡ 0 , 1 , − 1 mod (9)
Moreover since there are 5 cubes so therefore atleast one of them must be divisible by 9
Now if 9 ∣ a 1 3 + a 2 3 + a 3 3 + a 4 3 + a 5 3
Then there are equal no. of cubes which leave a remainder 1 and -1. but since 5 is odd one will leave a remainder 0
So since the cube is divisible by 9 the no. is divisible by 3