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Thirty Six Exactly

The number 12 = 2^2 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

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Even So

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

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Power Crazy

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

Lastly - Well

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
How can you simplify this so that it does not contain a power raised to a power?

Can you find a pattern in the last two digits? How about starting with some simple cases?