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.