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14 Divisors

What is the smallest number with exactly 14 divisors?

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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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Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

Largest Product

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

This problem can be tackled at very different intellectual levels. It is an excellent context to practise adding and multiplying with decimals while trying to solve a problem.

Part of the problem is to realise that there is, in fact, a question to consider.

Possible approach

Ask the students for various sets of numbers that add to 10 (exclude negatives!). Ask students to work out the products of some of these sets - students might demonstrate the methods on the board to help the group remember the algorithms.

Present the problem for the lesson - to find the biggest product, for numbers that add to $10$, to keep a sensible record of things that they have tried and what did and didn't work out.

After some time working, students could feed back on successful strategies they are using, and the class 'best' could be put up on the board for others to beat. Make sure that students know that there are ways of adapting the problem if they feel that they are making no further progress.

Key questions

  • How shall we start to make some progress with this challenge ?

  • Can you generalise your results somehow?

  • What is the same and what is different about the solution if the $10$ is changed to another number?

Possible extension

What about the sum being $0.5$ or $1.1$ or $-8$ etc., how would the strategy/solution change?

Students often work by reasoning from a few 'spot' values and showing that improvement occurs as they adjust towards a particular value, which they therefore declare to be the optimum. It is certainly good to draw attention to the logical possibility that an even better 'local maximum' might exist somewhere else. And also, if they are up to it, that adjustments are always incremental so how can we be sure that we haven't jumped right over an interval which contains something important.
Can we reason that this function is continuous ?

Possible support

With students who aren't ready for working with these decimals, perhaps consider integer solutions only for various sums.