Question

Let $x,y,z$ be positive integers such that $HCF(x,y,z)=1$ and $x^2+y^2=2z^2$. Which of the following statements are true?
I. $4$ divides $x$ or $4$ divides $y$.
II. $3$ divides $x+y$ or $3$ divides $x-y$.
III. $5$ divides $z(x^2-y^2)$.

• A. I and II only
• B. II and III only
• C. II only
• D. III only

Answer 1

Answer: (B)

II and III only

If H.C.F$(x,y,z)=1$, then $x,y,z$ are co-prime numbers and given that $2z^2$ is sum of the squres of $2$ co-prime numbers.

Therefore, possible value of $z=5⟹2z^2=50={49}^2+1^2$

$x^2+y^2=49+1=7^2+1^2$

Thus, $x=7,y=1,z=5$ or $x=1,y=7,z=5$ are possible values.

$x-y$ is divisible by $3$ and $z(x^2-y^2)$ is divisible by $5$.

Statement || and ||| are true.