Three positive integers are equally spaced on a number line. The middle number is $15,$ and the largest number is $4$ times the smallest number. What is the smallest of these three numbers?
Let the smallest number be $x.$ It follows that the largest number is $4x.$ Since $x,15,$ and $4x$ are equally spaced on a number line, we have \begin{align} 4x-15 &= 15-x \\ 5x &= 30 \\ x &= \fbox{6}. \end{align}