Sum of Multiples in Interval

The best way to illustrate how to find the Sum of Multiples in an Interval is with an example. Imagine we want to find the sum of all multiples of $3$ from $1$ to $500$. Follow these steps:

• Step 1: Find the number of multiples of $3$ from $1$ to $500$.
  • $\frac{500}{3}$, take the whole number result, to get 166. Put this number aside for now.
• Step 2: List out the first three multiples and last three multiples in the interval.
  • $3$, $6$, $9$...$492$, $495$, $498$
• Step 3: Notice that every "pair" in that list above adds up to $501$.
  • $3+498=501$
  • $6+495=501$
  • $9+492=501$
• Step 4: We know that every "pair" in the list adds up to $501$. We also know that we have a total of $166$ multiples of $3$ (from step 1), giving us $83$ pairs ($\frac{166}{2}$).  We can then find the total sum by multiplying these two numbers together (see below):

$$\frac{166}{2} \times 501 = 41,583$$

Pretty frickin' cool if I must say so myself.

NOTE: This trick still works even if you find an odd number of multiples in an interval. For example, there are $9$ multiples of $3$ from $1$ to $28$ and each pair adds up to $30$. To find the sum of all multiples, see below:

$$\frac{9}{2} \times 30=135$$