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Let $M$ be the set of all $a=1+4k$, $k\geq 0$. If $a, ab \in M$ then $b$ is in $M$. It's probably really easy, I just need a hint. Thank you.

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A Mersenne number has the form $2^n - 1$, for $n$ a positive integer (usually prime). Why did you mention these in your title? It seems out of place. –  hardmath Aug 15 at 10:16

3 Answers 3

Hint:

$\left(1+4k\right)b=1+4m\iff b=1+4\left(m-kb\right)$

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+1 for this surprisingly direct method. –  Hagen von Eitzen Aug 15 at 11:02

Hint:

All integers are of one of this forms:

$$4k \hspace{1cm} 4k+1 \hspace{1cm} 4k+2 \hspace{1cm} 4k+3$$

Multiply those by $4k'+1$ and see wich cases lead to an element of $M$.

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thanks a lot :-) –  Jana Aug 15 at 10:02

Suppose $b∉M$ then $b$ is one of the form $4k, 4k+2, 4k+3.$ This would be help for you.

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thank you very much –  Jana Aug 15 at 10:01

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