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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?

One or Both

Age 11 to 14 Challenge Level:

An ideal opportunity to discuss different approaches to a solution which might result from students working on the first part of the problem.

Out of the methods adopted by the students might emerge the idea that the pupils in the problem who got both questions correct have been counted twice so 9 must represent any percentage over 100. This might be a useful idea when trying to tackle the second part of the problem.

The third part of the problem aims at encouraging greater generalisation and therefore reinforce some of the ideas developed in the first part of the problem.