MHB Integer solutions of system of equations

AI Thread Summary
The discussion focuses on finding all integer solutions for the equations x+y+z=3 and x^3+y^3+z^3=3. Participants are attempting to clarify and correct previous answers, indicating that there may have been misunderstandings or errors in earlier responses. The conversation highlights the importance of accuracy in mathematical solutions and the need for clear communication. The initial inquiry remains unresolved as participants seek the correct integer solutions. The thread emphasizes collaborative problem-solving in mathematics.
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Find all integer solutions of the system of equations $x+y+z=3$ and $x^3+y^3+z^3=3$.
 
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we are given
$x+y+z = 3 \cdots(1)$
and
$x^3+y^3+z^3 = 3\cdots(2)$
from (1)
$x+y = 3 - z\cdots(3)$
and from (2)
$x^3+y^3 = 3 - z^3\cdots(4)$
From (3) and (4)
because $x+y$ divides $x^3+y^3$ so $x+y$ divides $3-z^3$ or $3-z$ divides $3-z^3$
so $z-3$ divides $z^3- 3$
as $z-3$ divides $z^3-3^3$ or $z^3 - 27$
so $z-3$ divides $(z^3-3) - (z^3- 27) = 24$
further if we have mod 9 then
$x^3 = 0\, or 1\,or\, -1$
$y^3 = 0\, or 1\,or\, -1$
$z^3 = 0\, or 1\,or\, -1$
as we have $x^3+y^3+z^3 = 3$ so we have $x^3=y^3=z^3 = 1$ mod 9
so $x \equiv y \equiv z \equiv 1\pmod 3$
so we need to take x-3 such that they are 1 mod 3 and factor of 24
they are ${ -8, -2, 1, 4}$
This gives choices for x as $(-5, 1, 4, 7)$
same for y and z and we can checking the sets get $x=y=z=1$
 
Last edited:
Sorry kaliprasad, your answer is not quite right...
 
anemone said:
Sorry kaliprasad, your answer is not quite right...
May be. I would like to know the correct answer
 
There was a typo error in first line and I corrected the same. Otherwise I do not find error if any. This may be pointed
 
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