### 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:

Powers of numbers behave in surprising ways...

Take a look at the following and try to explain what's going on.

Work out $2^1, 2^2, 2^3, 2^4, 2^5, 2^6$...
For which values of $n$ will $2^n$ be a multiple of $10?$

For which values of $n$ is $1^n + 2^n + 3^n$ even?

Work out $1^n + 2^n + 3^n + 4^n$ for some different values of $n$.
What do you notice?

What about $1^n + 2^n + 3^n + 4^n + 5^n?$

What other surprising results can you find?

Here are some suggestions to start you off:

$4^n + 5^n + 6^n$
$2^n+3^n$ for odd values of $n$
$3^n + 8^n$
$2^n + 4^n + 6^n$
$3^n + 5^n + 7^n$
$3^n - 2^n$
$7^n + 5^n - 3^n$