Let a,b,c be integers such that the average of the numbers 7a−9, 7b−9, 7c−9is2014.Prove that the average of the numbers a,b,c is a perfect square

Question

Let a,b,c be integers such that the average of the numbers 7a−9, 7b−9, 7c−9is2014.Prove that the average of the numbers a,b,c  is a perfect square

ganeshie8 · Answer

ganeshie8 · Answer

kids problem :P

anonymous · Answer

:P
well assume you dnt know the avg solve it

anonymous · Answer

lets say solve it theoritical

ganeshie8 · Answer

if i dnt even knw avg, how would u expect me knw theorotical lol

anonymous · Answer

humm number theory :O

anonymous · Answer

im trying lol , found it interested :3

ganeshie8 · Answer

7a−9 + 7b−9 + 7c−9 = 3*2014

ganeshie8 · Answer

is that the starting step ?

anonymous · Answer

idk xD

anonymous · Answer

so it means a+b+c/3 should be of the form n^2

ganeshie8 · Answer

yep

anonymous · Answer

nvm lol , i wake up suddly thinking of this xD
ill go back to slp

ganeshie8 · Answer

:) i thought u had some nice way to solve this... without using average thing... .

anonymous · Answer

unless 
7(a+b+c)-9 /3 = 2014 
its weird cuz
2014=7(..)+5 :-\

ganeshie8 · Answer

you get
 (7(a+b+c)-27)/3 = 2014

right ?

anonymous · Answer

O.O 
yeah

ganeshie8 · Answer

(7(a+b+c)-27)/3 = 2014

7(a+b+c)/3 - 9 = 2014

7(a+b+c)/3 = 2023

(a+b+c)/3 = 289 = 17^2

QED, go back to slp :D

anonymous · Answer

haha ok good night

anonymous · Answer

btw im not sure if i could sleep peacefully until i figure out something else :-\|dw:1400189678388:dw|
but its better to slp :3