Math Help - Proof help

1. Proof help

For all integers a and b, if a^2 is congruent to b^2 (mod 20), then a is congruent to b (mod 5). prove true or false

2. Re: Proof help

Originally Posted by lemondonut
For all integers a and b, if a^2 is congruent to b^2 (mod 20), then a is congruent to b (mod 5). prove true or false
If If $a^{2} \equiv b^{2}\ \text{mod}\ 20$, then there is a constant k for which is...

$(a+b)\ (a-b)= 20\ k$ (1)

From (1) is we derive that must be $5|(a+b)$ or $5|(a-b)$. If $5|(a-b)$ then is $a \equiv b\ \text{mod}\ 5$. If $5|(a+b)$ then for (1) it must be...

$(a-b)\ \frac{a+b}{5}= 4\ k \implies a-b= \frac{5}{a+b}\ 4\ k$ (2)

... so that $5|(a-b)$...

Kind regards

$\chi$ $\sigma$