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Counting Factors

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

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Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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Helen's Conjecture

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

Even So

Age 11 to 14 Challenge Level:

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?