# Thread: Modulo of squares = modulo of roots

1. ## Modulo of squares = modulo of roots

Stuck again! By the way thanks again for all the help!

Prove or disprove:

For all positive integers a, b: if $a^2$ $\equiv$ $b^2$ (mod 9), then a $\equiv$ b (mod 9)

I haven't found a counterexample so it looks true and I have the proof for the converse. I've only gotten this far:

Given
$a^2$ $\equiv$ $b^2$ (mod 9)

then 9 | ( $a^2$ - $b^2$) = (a - b)(a + b)

I know that if s | tu then s | t or s | u.
So 9 | (a - b) or 9 | (a + b) and we need to show that 9 | (a - b) but I'm not sure where to go from here.

Thanks

2. Originally Posted by oldguynewstudent
Stuck again! By the way thanks again for all the help!

Prove or disprove:

For all positive integers a, b: if $a^2$ $\equiv$ $b^2$ (mod 9), then a $\equiv$ b (mod 9)

I haven't found a counterexample so it looks true and I have the proof for the converse. I've only gotten this far:

Given
$a^2$ $\equiv$ $b^2$ (mod 9)

then 9 | ( $a^2$ - $b^2$) = (a - b)(a + b)

I know that if s | tu then s | t or s | u.
So 9 | (a - b) or 9 | (a + b) and we need to show that 9 | (a - b) but I'm not sure where to go from here.

Thanks
Surely 1 and, for example, 8 work...

3. Originally Posted by oldguynewstudent
Stuck again! By the way thanks again for all the help!

Prove or disprove:

For all positive integers a, b: if $a^2$ $\equiv$ $b^2$ (mod 9), then a $\equiv$ b (mod 9)

I haven't found a counterexample so it looks true and I have the proof for the converse. I've only gotten this far:

Given
$a^2$ $\equiv$ $b^2$ (mod 9)

then 9 | ( $a^2$ - $b^2$) = (a - b)(a + b)

I know that if s | tu then s | t or s | u.
So 9 | (a - b) or 9 | (a + b) and we need to show that 9 | (a - b) but I'm not sure where to go from here.

Thanks
"I know that if s | tu then s | t or s | u." - This is wrong - and rather a grave mistake. I will let you think about it.

4. Hello oldguynewstudent

Following the previous replies, you might also like to work out why the following are also counterexamples:
$a=2, b= 7$

$a=3,b=6$

$a=4, b=5$