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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 to Eight

Age 11 to 14 Challenge Level:

Here are some 'funny factorisations'. Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits $2$ to $9$ once and only once.

** $\times$ ** $= 4876$
** $\times$ ** $= 5394$

Now do the same with the digits $1$ to $8$ to complete the following expressions.

** $\times$ ** $= 1368$
$5$* $\times$ $6$* $=$ ****
$52$ $\times$ ** $=$ ****

Is there more than one solution?