Yes, this is true.

Indeed, by a result of Denis Hanson (Canad. Math. Bull., 1973) the product (I use notation to match the papers not the question) $$\Delta(n,k)= n (n+1) \dots (n+k-1)$$ for $n \ge k$ is divisible by a prime of size at least $3k/2$ with only the exception of $3\cdot 4$ , $8 \cdot 9$ and $6\cdot 7 \dots 10$.

There are also further results in this direction. For example Laishram and Shorey (Acta Arith., 2005) proved that the largest prime divisor of $\Delta(n,k)$ is strictly greater $1.97k$ if $n \gt k+13$ and $2k$ if in addition $n> (279/262) k$

Yes, this is true.

Indeed, by a result of Denis Hanson (Canad. Math. Bull., 1973) the product (I use notation to match the papers not the question) $$\Delta(n,k)= n (n+1) \dots (n+k-1)$$ for $n \ge k$ is divisible by a prime of size greater $3k/2$ with only the exception of $3\cdot 4$ , $8 \cdot 9$ and $6\cdot 7 \dots 10$.

There are also further results in this direction. For example Laishram and Shorey (Acta Arith., 2005) proved that the largest prime divisor of $\Delta(n,k)$ is strictly greater $1.97k$ if $n \gt k+13$ and $2k$ if in addition $n> (279/262) k$