find the the greatest value of $m$ such that $\text{lcm}(1,2,3,..,n)=\text{lcm}(m,m+1,..,n).$ I am stuck and unable to proceed. Value of n can be very large. For eg: if $n=6,\ \text{lcm}(1,2,...,6)=60$, so answer will be $4$ in this case.  
Since $\text{lcm}(2,3,4,5,6)=60,\ \text{lcm}(3,4,5,6)=60,\ \text{lcm}(4,5,6)=60$ and $\text{lcm}(5,6)=30...$ so largest $m$ in this case will be $4$.   
So in the question I am given any value of $n$ in range $1$ to $10^9$  and I need to tell the largest value of $m$.

Let $L_n$ be the left-hand side, and let $R_{m,n}$ be the right-hand side.  Note that $L_n$ is divisible by all of the greatest prime powers $p^r \leq n$.  In fact, $$L_n= \prod_{p \leq n} p^r$$ where $p$ is prime and $r$ is the greatest exponent such that $p^r \leq n$.  
Clearly $R_{m,n} \mid L_n$.  Therefore, $$L_n=R_{m,n} \iff p^r \mid R_{m,n}$$ for each prime $p \leq n$ and greatest $r$.  
Set $m_p=\max\{kp^r|1 \leq k \leq n/p^r\}$.  Then $m_p \leq n$ is the largest value divisible by $p^r$.  Finally, we know $$m=\min\{m_p | p \leq n\}$$
