A9. Dividing 52 by 12 gives 4 remainder 4 . What is the sum of all the numbers for which dividing by 12 gives a whole number answer which is the same as the remainder?
SolUTION
858

Using the example given we can write $52=12 \times 4+4=13 \times 4$, and so in general we are looking for numbers of the form $12 m+m=13 m$ where $m$ is any possible remainder when dividing by twelve. When dividing by twelve the possible remainders are $0,1,2, \ldots, 11$. The sum we want is $13 \times 0+13 \times 1+13 \times 2+\ldots+13 \times 11$. This can be rewritten as $13(0+1+2+\ldots 11)=13 \times 66$ and so the desired sum is 858 .