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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?

Powerful Zeros

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

Imagine the number $3^4\times 4^5\times 5^6$ is written out in full. How many zeros are there at the end of the number?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.
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