2005 AMC 10A Problem 21

Since $1 + 2 + \cdots + n = \dfrac{n(n+1)}{2},$ the quotient is $\dfrac{6n}{\frac{n(n+1)}{2}} = \dfrac{12}{n+1},$ which is an integer exactly when $n + 1$ divides $12.$ The divisors of $12$ that are at least $2$ are $2,$ $3,$ $4,$ $6,$ $12,$ giving $n = 1,$ $2,$ $3,$ $5,$ $11$ — five values. Thus, the correct answer is B.