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5x = 0 (mod 6)

I don't even know where to begin.

Ok, does 0(mod6) = 0mod6?

0/6 always give 0 so is the answer x=0?

I don't have any clue.

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An answer is $0$, but note that $5\times 6 = 30 \equiv 0 \pmod 6$ so $6$ is also a solution and there are many more. – Henry Oct 27 '12 at 20:28
Do you know what $a=b \pmod{n}$ means? – Chris Eagle Oct 27 '12 at 20:35
up vote 3 down vote accepted

Recall that if $a \vert (bc)$ and $(a,b)=1$, then $a \vert c$.

In your case, note that $6 \vert (5x)$ and $(6,5) = 1$. Hence, $6 \vert x$ i.e. $x \equiv 0 \pmod 6$

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Should the symbol be ≡ instead of =? – Penny Loren Oct 27 '12 at 20:42

Hint $\rm\,\ mod\ 6\!:\,\ 6x,5x\equiv 0\:\Rightarrow\:x = 6x\!-\!5x\equiv 0,\:$ and conversely.

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