### Digit Sum

What is the sum of all the digits in all the integers from one to one million?

### Big Powers

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

# Repeaters

##### Age 11 to 14 Challenge Level:

Tim from Gravesend Grammar School and Mohammad Afzaal Butt both sent us similar solutions to the problem. Well done Tim and Mohammad. Here is Mohammad's solution:

Let the three digit number be $xyz$. Hence the six digit number will be $xyzxyz$. Now
\eqalign { xyzxyz &= 100000x + 10000y + 1000z + 100x + 10y + z \cr &= 100100x + 10010y + 1001z \cr &= 1001 (100x + 10y + z) \cr &= 7 \times 11 \times 13 (100x + 10y + z)} Hence the number $xyzxyz$ is always divisible by $7$, $11$ and $13$.