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Let $p\ge 5$ be a fixed prime number and let $S=\{2, 3, \ldots, p\}$ be the set of prime numbers between $2$ and $p$.

Which of the following is FALSE?

$N=(\prod_{q\in S}q)+1$ is not divisible by any element of $S$.

$M=2^2(\prod_{q\in S, q\not=2}q)-1$ is of the form $4n+3$, for some integer $n$.

$T=(\prod_{q\in S}q)-1$ is of the form $6n+5$, for some integer $n$.

If $m\ge 1$ is an integer, and $2^m+1$ is prime, then $m$ is a power of $2$.

If $m\ge 4$ is an integer, and $2^m-1$ is prime, then $m$ is a power of $2$.