### 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.

# Missing Digit

##### Stage: 3 Short Challenge Level:
A test for divisibility by 11 is to add alternate digits:

1 + 3 + * + 7 = 11 + *; 2 + 4 + 6 + 8 = 20.

If the original number is a multiple of 11 then these two totals will be the same or will differ by a multiple of 11. In this case, 11 + * = 20 gives * = 9.

Or, you can solve it without knowing a rule as follows:

1234*678 = 12340678 + 1000* = (11 x 1121879 +9) + 11 x 90* + 10*

and hence is divisible by 11 if and only if 10* + 9 is divisible by 11. So * = 9.

This problem is taken from the UKMT Mathematical Challenges.