Let x, y, z be positive integers such that HCF (x, y, z)=1 and x^2+y^2=2 z^2 . Which of following st

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4-2.Quadratic Equations and Inequations

normal

Let $x, y, z$ be positive integers such that $HCF$ $(x, y, z)=1$ and $x^2+y^2=2 z^2$. 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 $z\left(x^2-y^2\right)$.

$I$ and $II$ only

$II$ and $III$ only

$II$ only

$III$ only

(KVPY-2017)

Solution

(b)

We have

$\quad x^2+y^2=2 z^2 \text { and HCF }(x, y, z)=1$

$x=1, y=7, z=5$

$\text { Then, } \quad 1+49=50$

$\text { or } \quad x=7, y=1, z=5$

$\quad 49+1=50$

$\therefore 4 \text { not divides } x \text { or } y.$

$\therefore \text { Ist statement is wrong. }$

$\quad x+y=7+1=8$

$\quad x-y=7-1=6$

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Let $x, y, z$ be positive integers such that $HCF$ $(x, y, z)=1$ and $x^2+y^2=2 z^2$. 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 $z\left(x^2-y^2\right)$.