For how many positive values of $n$ are both $\frac n3$ and $3n$ four-digit integers? For how many positive values of $n$ are both $\frac n3$ and $3n$ four-digit integers?
Any help is greatly appreciated. I think the smallest n value is 3000 and the largest n value is 3333. Does this make sense?

The question is asking how many values of n satisfy 


*

*$n \in \mathbb{N}$

*$3n \in [1000, 9999]$ 

*$\frac n3 \in [1000, 9999]$


So find the min value of $n$ satisfying the lower bound:
$$ \frac n3 \ge 1000 $$
And the max value of $n$ satisfying the higher bound:
$$ 3n \le 9999$$
Now only include values in the range $[3000, 3333]$ where $\frac n3$ is an integer (adding 1 since $3000$ is divisible by $3$).
$$ \frac {3333}{3} - \frac {3000}{3} + 1 = 112 $$
