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### Number and algebra

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# Power Mad!

## You may also like

### Adding All Nine

### Repeaters

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Links to the NRICH website Home page

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30 April (Primary), 1 May (Secondary)

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Age 11 to 14

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 $n$

$3^n + 8^n$

$2^n + 4^n + 6^n$

$3^n + 5^n + 7^n$

$3^n - 2^n$

$7^n + 5^n - 3^n$

**Can you justify your findings?**

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

Click here for a poster of this problem.

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

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.