David Speyer has given a very short proof of an upper bound for $M(n)$ in the comments. There is a similarly short argument that provides a lower bound on $M(n)$ which I believe is due to Gelfond and Schnirelmann, which goes as follows:

Let $P_n(x)$ denote a degree $n-1$ polynomial with integer coefficients. Then clearly the product $M(n) \int_{0}^{1} P_n(x) dx$ is an integer. Now consider $P(x) = x^{\frac{n-1}{2}}(1-x)^{\frac{n-1}{2}}$. For $x \in [0,1]$ we have that $P_n(x) < 2^{-(n+1)}$ since the maximum of $x(1-x)$ occurs at $x=1/2$. Thus $1 \leq M(n) \int_{0}^{1} P_n(x) dx < 2^{-(n-1)}$. Rearranging things we have that $M(n) > 2^{n-1}.$

David Speyer has given a very short proof of an upper bound for $M(n)$ in the comments. There is a similarly short argument that provides a lower bound on $M(n)$ which I believe is due to Gelfond and Schnirelmann, which goes as follows:

Let $P_n(x)$ denote a degree $n-1$ polynomial with integer coefficients. Then clearly the product $M(n) \int_{0}^{1} P_n(x) dx$ is an integer. Now consider $P(x) = x^{\frac{n-1}{2}}(1-x)^{\frac{n-1}{2}}$. For $x \in [0,1]$ we have that $P_n(x) \leq 2^{-(n-1)}$ since the maximum of $x(1-x)$ occurs at $x=1/2$. Thus $1 \leq M(n) \int_{0}^{1} P_n(x) dx < M(n) 2^{-(n-1)}$. Rearranging things we have that $M(n) > 2^{n-1}.$

David Speyer has given a very short proof of an upper bound for $M(n)$ in the comments. There is a similarly short argument that provides a lower bound on $M(n)$ which I believe is due to Gelfond, which goes as follows:

Let $P_n(x)$ denote a degree $n-1$ polynomial with integer coefficients. Then clearly the product $M(n) \int_{0}^{1} P_n(x) dx$ is an integer. Now consider $P(x) = x^{n-1} (1-x)^{n-1}$. For $x \in [0,1]$ we have that $P_n(x) < 4^{-(n+1)}$ since the maximum of $x(1-x)$ occurs at $x=1/2$. Thus $1 \leq M(n) \int_{0}^{1} P_n(x) dx < 4^{-(n-1)}$. Rearranging things we have that $M(n) > 4^{n-1}.$

David Speyer has given a very short proof of an upper bound for $M(n)$ in the comments. There is a similarly short argument that provides a lower bound on $M(n)$ which I believe is due to Gelfond and Schnirelmann, which goes as follows:

Let $P_n(x)$ denote a degree $n-1$ polynomial with integer coefficients. Then clearly the product $M(n) \int_{0}^{1} P_n(x) dx$ is an integer. Now consider $P(x) = x^{\frac{n-1}{2}}(1-x)^{\frac{n-1}{2}}$. For $x \in [0,1]$ we have that $P_n(x) < 2^{-(n+1)}$ since the maximum of $x(1-x)$ occurs at $x=1/2$. Thus $1 \leq M(n) \int_{0}^{1} P_n(x) dx < 2^{-(n-1)}$. Rearranging things we have that $M(n) > 2^{n-1}.$