### Counting Factors

Is there an efficient way to work out how many factors a large number has?

### 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.

##### Stage: 3 Challenge Level:

List any 3 numbers.

It is always possible to find a subset of adjacent numbers that add up to a multiple of 3 (that is either one, two or three numbers that are next to each other). For example:

 5, 7 , 1 5 + 7 = 12 (a multiple of 3) 4,4, 15 15 is a multiple of 3 5,11,2 5 + 11 + 2 = 18 (a multiple of 3)

Can you explain why and prove it?

What happens if you write a list of 4 numbers?
Is it always possible to find a subset of adjacent numbers that add up to a multiple of 4?
Can you explain why and prove it?

What happens if you write a long list of numbers (say n numbers)?
Is it always possible to find a subset of adjacent numbers that add up to a multiple of $n$?
Can you explain why and prove it?