Prove or disprove $a^2\mid b^3\Longrightarrow a\mid b$

Asked Mar 03 '16 at 20:15

Active Mar 03 '16 at 20:17

Viewed 97 times

I need to prove or to give a counter-example:

$$a^2\mid b^3\Longrightarrow a\mid b$$

My attempt:

First, let's check with small integres,trying to find counter-example:

$2^2\mid 6^3\Longrightarrow 2\mid 6\\ 1^2\mid 2^3\Longrightarrow 1\mid 2$

it looks correct,let's try to prove it:

$$a^2\mid b^3\Longrightarrow k \cdot a^2=b^3\Longrightarrow k \cdot aa=bbb\\$$

here I'm stuck

I don't know if this and this related because here it's different

edited Apr 13 '17 at 12:20

Community

asked Mar 03 '16 at 20:15

Error 404

11

Hint: what would happen if you had $a^2=b^3$? – Wojowu Mar 03 '16 at 20:16
@Wojowu then $a,b=1$ or $a,b=0$? – Error 404 Mar 03 '16 at 20:18
1

Nope, there actually are nontrivial examples. Try to look for one. – Wojowu Mar 03 '16 at 20:19
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@Wojowu $a=27$ and $b=9$ – Error 404 Mar 03 '16 at 20:26
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What does it tell you about the statement you tried to prove? – Wojowu Mar 03 '16 at 20:27
Ok, I got it thank you – Error 404 Mar 03 '16 at 20:27
Or $a=8$, $b=4$, for an example with smaller numbers.. – MathematicianByMistake Mar 03 '16 at 20:46
Or $a=1000$ and $b=100$, then both $a^2$ and $b^3$ is $1000000$, but $a$ is greater than $b$, hence cannot divide it. – Jeppe Stig Nielsen Feb 10 '18 at 12:09

0 Answers0