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N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

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Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Power Mad!

Stage: 3 and 4 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.