Question

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

• A. $c^2 \equiv a^2(mod\, b^2)$
• B. $a^2 \equiv b^2(mod\, c^2)$
• C. $a^2+b^2-b^2$
• D. $b^2 \equiv c^2(mod \, a^2)$

Answer 1

The Correct Option is D

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}} $
$\Rightarrow a^{2} \mid\left(b^{2}-c^{2}\right)$
$\Rightarrow b^{2} \equiv c^{2}\left(\bmod a^{2}\right)$