You may also like

problem icon

Counting Factors

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

problem icon

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.

problem icon

Oh! Hidden Inside?

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

Power Mad!

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Powers of numbers behave in surprising ways...

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


This power pylon is unlikely to help...
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 $n$
$3^n + 8^n$
$2^n + 4^n + 6^n$
$3^n + 5^n + 7^n$
$3^n  - 2^n$
$7^n + 5^n - 3^n$

Can you justify your findings?

You may also like to take a look at Big Powers.


Click here for a poster of this problem.