Find the sum of all the odd numbers between 10 and 550 that are divisible by three.

Answered July 28 by

**Mike Alexander**Studied A-Level Mathematics, Physics & Computer Science

So, the odd numbers in the range go: [he shortens lists for visual reasons]

11 13

**15**17 19

**21**23 25

**27**…

**543**545 547

**549**

Every third number

**in bold**is divisible by 3.

[Remember the math fact: if the sum of the digits in the number equals three, the number is divisible by three. Try it with each one of the boldface numbers.]

So, the sum is

15+21+27+33 … +543+549

Since all terms are divisible by 3, we can write this as:

3(5+7+9+11 … +181+183)

[Because multiplication is the inverse of division. Proof: multiply the last number, 183, by three, and the product is 549.]

Now notice that the series inside the parentheses is almost a full set of consecutive odd numbers - it just lacks 1+3+ at the start. So, it will be 4 less than the series:

I will stop here. If you want the rest of the proof, click https://www.quora.com/How-do-I-do-this-arithmetic-question-Find-the-sum-of-all-the-odd-numbers-between-10-and-550-that-are-divisible-by-3

The final answer is 25,380.

Guess what? The answer is divisible by three.