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Let $Z_8$ be the set of all congruence classes of integers $\text {modulo} \ 8$. Find positive integers $a,b$ such that $[a][b]=[0]$. ($[a]$ denotes the congruence class of $a$ $\text {modulo}\ 8$

First of all, I have no idea what does $[b]$ denote.

If $$[a][b]=[0]\\ a \mod 8. [b]=[0]$$. But here I am clueless and stuck.

Now if it wants to know for which positive $a,b$, we can represent $a.b\equiv0\mod 8$, as any or both of them are even. Otherwise I don't know how to solve it.

I need help to solve and any help is highly appreciated.

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  • $\begingroup$ $[a]$ is the remainder of $a$ on division by $8,$ considered as an element of $\mathbf Z_8.$ For example, $[33]=1\in \mathbf Z_8.$ $\endgroup$ – saulspatz Mar 11 '18 at 20:37
  • $\begingroup$ As your exercise says, $[b]$ is the congruence class mod $8$ that contains the integer $b$. $\endgroup$ – hardmath Mar 11 '18 at 20:39
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One example could be $$[a]\equiv 2\pmod 8 \quad\text{and}\quad [b]\equiv 4\pmod 8$$ You are looking for the zero divisors, and in $\mathbb{Z}_8$ they are the numbers that are not coprime with $8$.

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  • $\begingroup$ Is it similar to $a.b\equiv0\mod 8$ ? $\endgroup$ – thevbm Mar 11 '18 at 20:40
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    $\begingroup$ @thevbm if you mean $a\color{red}{\cdot}b\equiv 0\pmod 8$ you are right! $\endgroup$ – user507623 Mar 11 '18 at 20:42
  • $\begingroup$ When we talk about congruences we always deal with infinite number of solutions. In your case all the solution are of the form $[a][b]=8k\quad\forall k\in\mathbb{Z}_8$. $\endgroup$ – user507623 Mar 11 '18 at 20:51
  • $\begingroup$ So for example $8.2=2.8=4.4=[a].[b]$ or $[a][b]=1.8=8.1=2.4=4.2=8.k$ am I correct? $\endgroup$ – thevbm Mar 11 '18 at 20:55
  • $\begingroup$ Yes. But remember that $[8]\equiv [0]\pmod 8$ $\endgroup$ – user507623 Mar 11 '18 at 21:14

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