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 the numbers from $1$ to $5$?

Answer 1

To find the rotcaf of a number, we need to find the greatest factor of the number that is less than the number and add it to the number itself. We could list all the factors of each number, but instead, we will use the number 9 to find a pattern and then prove it works for all numbers. Since 1 is the only factor of 1, we will have $1+1$ for the rotcaf of $1$. Since 1 and 3 are both factors of 9 and 3 is the largest factor less than 9, we have $9+3$ for the rotcaf of $9$. Therefore, we can guess that a number has a rotcaf equal to itself plus the sum of all its factors excluding the number itself. For example, the factors of 9 excluding 9 itself are 1 and 3, so $9+3$ is the rotcaf of 9. Using 1 and 5 as factors of 10 excluding 10 itself, $10+1+5$ is equal to $16$, which is the rotcaf of $10$. Using this method, we find that the rotcaf of $1$ is $2$, the rotcaf of $2$ is $3$, the rotcaf of $3$ is $4$, the rotcaf of $4$ is $7$, the rotcaf of $5$ is $6$, and the sum of all rotcafs is $2+3+4+7+6=\boxed{22}$.