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So I need to write down the smallest natural number $x$ that satisfies this statement:

$11x$ has reminder of $1$, when divided by $2,3,5,7$.

So I know it's somehow solved by using modular arithmetic but I wonder how can I apply it here.

Any help would be appreciated.

Jean Marie
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MathIsTheWayOfLife
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    It looks as if $11x$ has a remainder of $1$ when divided by $210$ – Henry Feb 09 '17 at 11:36
  • Essentially you need to solve $11x+210y=1$ in integers. See https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm – Wolfram Feb 09 '17 at 11:38
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    Now note that $210=11 \times 19 +1$ and so $11\times (210-19) = 11^2 \times 19 +11-11 \times 19 +1$ – Henry Feb 09 '17 at 11:39

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