Finding the sum of x
Number Theory
Level
2
Let x and y be integers such that x and y are the solution of the equation 7x^4 - 5y^3 = 567(for example x=3 and y=0 is a solution). Find the sum of all possible value of x.
8
27
3
0
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1 solution
If ( x , y ) = ( x 0 , y 0 ) is a solution, meaning that 7 x 0 4 − 5 y 0 3 = 5 6 7 , then ( x , y ) = ( − x 0 , y 0 ) is also a solution because 7 ( − x 0 ) 4 − 5 y 0 3 = 7 x 0 4 − 5 y 3 .
This means that the sum of those two x-values is x 0 + ( − x 0 ) = 0 and therefore the sum of all x-values is also 0 .