## ganeshie8 one year ago If $$d^2$$ divides $$a^2$$, show that $$d$$ divides $$a$$ $$d,a\in \mathbb{Z}$$

1. zzr0ck3r
2. anonymous

that ruins the fun lol

3. zzr0ck3r

then don't look:) That is why I did not regurgitate the proof. I know I have seen it in NT but if someone has another way...

4. ganeshie8

It doesn't, you still can have your own proof :) it seems some proofs in that link are circular..

5. zzr0ck3r

I only looked at the first one

6. zzr0ck3r

I buy everything in the one with 49 votes.

7. zzr0ck3r

the exponent thing is weird, but great frigging idea

8. anonymous

simple let take a=1 b=2 now b^2/a^2=2^2/1^2 now take square root on both sides b/a=2/1

9. zzr0ck3r

this only shows it for one case, not all :)

10. zzr0ck3r

that would be like saying take $$4$$, well $$\sqrt{4}\in \mathbb{Z}$$ so it must be that $$\sqrt{x}\in \mathbb{Z}$$ for all real numbers $$x$$.

11. anonymous

we have to consider those numbers whose square root will be integer not in decimal form dude

12. zzr0ck3r

1.0

13. anonymous

means by dividing the numbers we get a number with complete square root

14. zzr0ck3r

you are correct in saying the theorem is true, we are saying that you have not sufficiently proved it. You should look at one of the proofs on the page I posted to see what the formal proof would look like.

15. anonymous

$$\Large \begin{matrix} \\ d = \prod_{i=1}^{n}p_i^{k_i} \\ d^2 = \prod_{i=1}^{n}p_i^{2k_i} \\ d^2|a^2\Rightarrow a^2=sd^2=S\prod_{i=1}^{n}p_i^{2k_i} \end{matrix}$$ since a^2 is perfect square then sd^2 is also perfect square, now we need to prove $$\sqrt s\in N$$

16. anonymous

you can say that XD ..

17. anonymous

ok we just say $$\Large a^2=sd^2=\prod_{i=1}^{m}p_i^{g_i}.\prod_{i=1}^{n}p_i^{2k_i}$$ W.L.G wither m>n or else, lets interpret in different variable l=max{n,m} $$\Large a^2 =\prod_{i=1}^{l}p_i^{2f_i} ~~such~ that ~~ f_i<=k_i ~\forall i$$ $$\Large a =\prod_{i=1}^{l}p_i^{ f_i} = \prod_{i=1}^{n}p_i^{k_i} \times something$$

18. anonymous

suppose d has value 2 and a has value 4. After d2 divides a2, we get value 1/4. http://www.acalculator.com/math-calculators.html