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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.

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    It looks as if $11x$ has a remainder of $1$ when divided by $210$ – 2017-02-09
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    Essentially you need to solve $11x+210y=1$ in integers. See https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm – 2017-02-09
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    Now note that $210=11 \times 19 +1$ and so $11\times (210-19) = 11^2 \times 19 +11-11 \times 19 +1$ – 2017-02-09

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