what is the smallest value that x can have?

Rewrite the system of equations as
3u - y + 3z = x
4u + 4y - z = x
-u + 5y + 5z = x.
Note that the determinants
　　|3 -1 3|
Δ = |4 4 -1|
　　|-1 5 5|
　= [(3)(4)(5) + (-1)(-1)(-1) + (3)(4)(5)] - [(3)(4)(-1) + (3)(-1)(5) + (-1)(4)(5)]
　= 166,
　　|x -1 3|
Δu = |x 4 -1|
　　|x 5 5|
　= x{[(1)(4)(5) + (-1)(-1)(1) + (3)(1)(5)] - [(3)(4)(1) + (1)(-1)(5) + (-1)(1)(5)]}
　= 34x,
　　|3 x 3|
Δy = |4 x -1|
　　|-1 x 5|
　= x{[(3)(1)(5) + (1)(-1)(-1) + (3)(4)(1)] - [(3)(1)(-1) + (3)(-1)(1) + (1)(4)(5)]}
　= 14x,
　　|3 -1 x|
Δz = |4 4 x|
　　|-1 5 x|
　= x{[(3)(4)(1) + (-1)(1)(-1) + (1)(4)(5)] - [(1)(4)(-1) + (3)(1)(5) + (-1)(4)(1)]}
　= 26x.
For any fixed x, by Cramer's rule, the solutions are hence given by
(u, y, z) = (17x/83, 7x/83, 13x/83).
For the system to have positive integral solutions, x must be a multiple of 83.
Hence the smallest value of x is 83.