212)
x+y+z=0
x+y=-z
(x+y)3 = (-z)3
x3+y3+3xy(x+y) = -z3
x3+y3+3xy(-z) = -z3
x3+ y3 + z3 = 3xyz
proved
1>how many odered pairs of (m,n) of positive integer are solution to
4m + 2n = 1 ?
2>prove that if a +b + c =0
then a3+b3+c3 = 3abc
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7 Answers
21212)
BY observation,
(8,4), (12,3), (6,6) , (5,10)
there may be some other process , experts may know
112) 1 obvious thing is if m>8 then n<4.
So our domain gets restricted - that is we have for n=3, m=12.
Now 4<m\le 8 - thus we can check consequently for m=5,6,7.
1I think ,
SFFT -- > Simon ' s Favourite Factorising Technique
The given equation can be factorised to -
( m - 4 ) ( n - 2 ) = 8
Now , we have to find out the number of such pairs of numbers which when multiplied give 8 .
Clearly , 8 = 8 x 1 = 1 x 8 = 4 x 2 = 2 x 4
So for each pair , we will get a " m " , and a " n " .
So , the solution set is -
m = 12 , n = 3
m = 5 , n = 10
m = 8 , n = 4
m = 6 , n = 6 .