06/04/2020 (2011: National Sprint Round, Problem 10)

Q: A rotcaf is the number that results from adding a number and the greatest factor of the number that is less than the number. For example, the rotcaf of 9 is 9 + 3, or 12. What is the sum of the rotcafs of all even numbers between 1 and 25?

A: 234

We have to look for a pattern in the rotcafs of even numbers between 1 and 25. Let's start with 2 and work our way up:

The rotcaf of 2 is: 2 + 1 = 3

The rotcaf of 4 is: 4 + 2 = 6

The rotcaf of 6 is: 6 + 3 = 9

As you can see, the rotcafs are just multiples of 3. Because there are 12 even numbers between 1 and 25, we just add the first 12 multiples of 3 and get:

3 + 6 + 9 + ... + 36 = (36 + 3)(12)(1/2) = (39)(6) = 234