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Find all the integral solutions of x(cube) = 6y(cube) + 20z(cube)
Since there are three variables x,y,z.Hence we need three equations to solve them
I have asked for integral solutions not its exact value. If it was to be exact value then three equations were required but I need integral solutions.
Its evident that x is even. This means LHS is divisible by 8
Now if y is odd and z even, then RHS leaves remainder 6 on division by 8
If y is even and z is odd, then RHS leaves remainder 4
If both y and z are odd, then RHS leaves remainder 2.
Hence y and z are both even
Let x=2p, y=2q, z=2r. Then after cancellation we have
That means if (x,y,z) is a solution so is (p,q,r).
But this process can be continued indefinitely, which is absurd (Infinite Descent)
Hence no solutions are possible
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