The correct option is A It has no positive integer solutions.
we can assume that gcd(x,y,z,t)=1.
Adding up the equations yields7(x2+y2)=z2+t2.
The square residues modulo 7 are 0,1,2, and 4.
It is not difficult to see that the only pair of residues which add up to 0 modulo 7 is (0,0), hence z and t are divisible by 7. Setting z=7z1 and t=7t1 yields 7(x2+y2)=49(z21+t21) or x2+y2=7(z21+t21).
It follows that x and y are also divisible by 7. Contradicting the fact that gcd(x,y,z,t)=1.