+23 Find $1_6 + 2_6 + 3_6 + \cdots + 45_6$. Express your answer in base $6$.
+505 There is probably a way to do this without first converting to base 10, but I'm not super sure how to do it.
Anyway, first convert 45 from base 6 to base 10, which is \(5+4\cdot6=29\).
The sum of the integers 1 to 29 base 10 is \(\frac{(29)(30)}{2}=435\). In base 6 that would be equal to \(\boxed{2003}\), which is our answer.
