I am stuck with the following problem of number theory.A number $n$ is said to be square full if $p^2|n$ for some prime $p$.
Let $P(k)$ be the product of all primes $\leq k$.Then show that there exist $31\times 21$ integers $n$ with $1\leq n\leq P(31)^2$ such that $n+1,n+2,...,n+10$ are all square full and none of them are divisible by $31$.(Note that $31$ is the $11$ th prime in ascending order.)
I tried to do that with chinese remainder theorem on $n\equiv -1\pmod{2^2},n\equiv -2\pmod{3^2},...,n\equiv -10\pmod{{p_{10}}^2}$,but then I am stuck.Can someone give me some hint?