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Let $a$ and $n$ be positive integers. A sequence of $n$ consecutive integers $(a, a+1, a+2,...,a+(n-1))$ is called a Wolczuk of length $n$ if every integer in the sequence is divisible by some perfect square greater than $1$. Example: $(48, 49, 50)$ is Wolczuk of length $3$ since $2^2\mid48$, $7^2\mid49$, $5^2\mid50$.
Prove that for any positive integer $n$, there exist infinitely many Wolczuks of length $n$.

I would really appreciate any help, it would be great if you prove this by Modular Arithmetic.

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By the Chinese remainder theorem, there exists $a\in\mathbb Z$ with $$\left\{\begin{align*} a&\equiv0\pmod{2^2}\\ a&\equiv-1\pmod{3^2}\\ a&\equiv-2\pmod{5^2}\\ a&\equiv-3\pmod{7^2}\\ &\;\;\vdots\\ a&\equiv-(n-1)\pmod{p_n^2} \end{align*}\right.$$

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