### Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

### Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

### Like Powers

Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n$ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

# One O Five

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

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100 but not to tell you which number they have in mind. Then ask them to tell you the remainders when this number is divided by 3, when it is divided by 5 and when it is divided by 7. Now multiply the first remainder by 70, the second remainder by 21 and the third remainder by 15 and add the three answers together. Now subtract multiples of 105 from this total to give as small a positive whole number as possible. This will be the number your friend first thought of. Test this out a few times and explain why it works.