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Lastly - Well

What are the last two digits of 2^(2^2003)?

Counting Factors

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

Summing Consecutive Numbers

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

Cogs

Age 11 to 14 Challenge Level:

A and B are two interlocking cogwheels having p teeth and q teeth respectively where $$3\leq q\leq p \leq 12$$

One tooth on B is painted red. Find the values of p and q for which this red tooth on B contacts every gap on the cogwheel A as the wheels rotate.