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How can we solve $2x\equiv 18\ (\operatorname{mod} 50)$? I'm not sure what to do when the item being modded on the right is not $1$.

3 Answers 3

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Hint : you can rewrite this equation as $2x-18 = 50k$ for some $k\in \mathbb{Z}$

this is what the modular equation translates to. now solve for x in terms of k to get your solutions

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    Yes, thanks for pointing out the typo. For clarity will rewrite. "This is equivalent to $x\equiv 9\pmod{25}$."2012-12-17
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Hint (pretty huge, but whadda...):

$2x=18\pmod{50}\Longleftrightarrow 2x=18+50k\,\,,\,k\in\Bbb Z\Longleftrightarrow x=9+25k\Longleftrightarrow\ldots$

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This means that $2x-18=50k$ for some integer $k$. Hence $x=25k+9$.