If (a | (b + c)) and (a| (b-c)) where (a,b,c \in , N ) then | Mathematics | SolveeIt

Question

If $a | (b + c)$ and $a| (b-c)$ where $a,b,c \in \, N$ then

$c^2 \equiv a^2(mod\, b^2)$

$a^2 \equiv b^2(mod\, c^2)$

$a^2+b^2-b^2$

$b^2 \equiv c^2(mod \, a^2)$

Answer

$b^2 \equiv c^2(mod \, a^2)$

Explanation

Solution

Since $a |(b+c)$ and $a|(b-c)$
$\Rightarrow \frac{b+c}{a}$ and $\frac{b-c}{a}$
$\therefore \frac{b+c}{a} \cdot \frac{b-c}{a}=\frac{b^{2}-c^{2}}{a^{2}}$
$\Rightarrow a^{2} \mid\left(b^{2}-c^{2}\right)$
$\Rightarrow b^{2} \equiv c^{2}\left(\bmod a^{2}\right)$

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