### Even Up

Consider all of the five digit numbers which we can form using only the digits 2, 4, 6 and 8. If these numbers are arranged in ascending order, what is the 512th number?

### Double Digit

Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. Try lots of examples. What happens? Can you explain it?

### What an Odd Fact(or)

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

# 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 $13$.