Showing existence of 2 constants

I want to show there exists 2 constants $\displaystyle c_1,c_2> 1$ such that:

$\displaystyle c_1^n\leq z_n\leq c_2^n $ where $\displaystyle z_n=\mathrm{lcm}(1,\cdots,n)=\prod_{p\leq n}p^{b_p}}$,

where $\displaystyle b_p$ is the largest number such that $\displaystyle p^{b_p}\leq n$

I accomplished to show the following: (more or less trivial)

$\displaystyle z_{n+1}=z_n\cdot n$ as n is prime

$\displaystyle z_{n+1}=z_n\cdot p$ as $\displaystyle n+1=p^{b_p}$ (i.e a power of only one prime)

$\displaystyle z_{n+1}=z_n$ otherwise (i.e. n+1 has at least 2 prime-divisors)

I believe these are useful observations, but I'm more or less stuck here. I hope someone can give me a little push in the right direction here...