Prove that the system of equations x^ 2 + 6y^ 2 -z^ 2 & amp; 6x^ 2 + y^ 2 - t^ 2 & amp has no positive integer solutions.

Prove that the system of equations x^ 2 + 6y^ 2 -z^ 2 & amp; 6x^ …

MCQ

Prove that the system of equations
$\begin{cases}\text{x}^{2} + 6\text{y}^{2}-\text{z}^{2} & \text{amp};\\6\text{x}^{2} + \text{y}^{2} - \text{t}^{2} & \text{amp}\end{cases}$
has no positive integer solutions.

✓
It has no positive integer solutions.
B
It has positive integer solution.
C
The equations do not have any solutions.
D
The equations has negative integer solution.

✓

Answer

Correct option: A.

It has no positive integer solutions.

we can assume that $\operatorname{gcd}( x , y , z , t )=1$.
Adding up the equations yields $7\left( x ^2+ y ^2\right)= z ^2+ t ^2$.
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 o modulo $7$ is $(o, o)$, hence $z$ and $t$ are divisible by $7$ .
Setting $z=7 z_1$ and $t=7 t_1$ yields $7\left( x ^2+ y ^2\right)=49\left( z _1^2+ t _1^2\right)$ or $x ^2+ y ^2=7\left( z _1^2+ t _1^2\right)$
It follows that $x$ and $y$ are also divisible by $7 .$
Contradicting the fact that $\operatorname{gcd}( x , y , z , t )=1$.