The hunt for perfect numbers

Factor chains are easy to create. Start with any positive integer and sum its factors (excluding the number itself). This gives you the next number in the chain.

So, for example, 10 gives the chain

Number chain

as the factors of 10 are 1, 2, 5 and 10, and 1 + 2 + 5 gives 8, and so on.

I usually give students a few examples of chains to try out themselves, and then see what they notice and ask the following questions:

  1. Do all of the chains end in 1?
  2. Is the penultimate number always prime?
  3. Are the numbers in the chain always decreasing?
  4. What is the shortest chain possible?
  5. What is the longest chain you can find?

Have a go and see what you find.

For more on this investigation, see the 'Hunt for perfect numbers' blog.

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