Divisors

Investigation

In the last section, hopefully you found the factors on 14 to be 1, 2, 7 and 14. Out of these, those which are less than 14 are called proper factors. In this case, 1, 2 and 7 are the proper factors of 14.

Now add up the proper factors of 14, (i.e. 1 + 2 + 7) what number do you get?

Find the proper factors of that number (i.e. find the proper factors of 10) , add the proper factors up and repeat the step until you think there is a good reason to stop.

What number do you end with? Why is it a good reason to stop here?

If you start with the number 14, the sequence should be 14 → 10 → 8 → 7 → 1. Can you now find all the sequences from 1 to 30 as your starting numbers? You may wish to share the workload in groups.

How can present your findings to show neatly how each of the numbers maps to other numbers?

What interesting properties have you noticed? If you need some prompts, look at the questions below:

• What do most numbers end in?

• Which are the exceptions? What do those numbers end in?

• What do you notice about the numbers that end in 1? Hint: how many factors do they have? What do we call this type of numbers? Answer here.

• Which numbers have an odd number of factors or an even number of proper factors? Why? What do we call this type of numbers? Answer here.

• Most numbers map to smaller numbers but some map to larger numbers, what are they? These are called abundant numbers, click on the hyperlink to learn more about them.

• Which numbers map to themselves? These are called perfect numbers, click on the hyperlink to learn more about them.

Further Questions and Challenges

Can you think of strategies to extend the chains backwards?

• Show that nothing maps to 2.

• Show that nothing maps to 5.

• Show that the only number that maps to 3 is 4.

• Which numbers map to 6?

• Which numbers map to 13? How do you know that you have found them all?

• Find an odd number greater than 5 that nothing maps to.

• What is the smallest number that has two different numbers mapping to it?

• What is the smallest number that takes two steps to reach 1? What about three steps? Four steps?

• What is the longest chain you can make starting with a number less than 50?

• Are there any groups of numbers that form their own loop?

• Show that all multiples of 6 from 12 upwards are abundant.

Here is a tough question... When is the sum of two numbers a factor of their product?

Further Practice

One essential skill introduced in this lesson is "factors". Other skills you may also wish to recap/ learn about are "prime factorisation", "HCF" and "LCM". Practise these relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.

Transum

Here is a self-marked practice on transum to check your understanding.

Looking for something fun to play with your friend? Try this "Connect 4 Factors" game. The winner is the first to line up four numbers with a common factor. You can also try this "Threes and Fives" game where the objective is for players to attach a domino from their hand to one end of those already played so that the sum of the end dominoes is divisible by five or three.

For more goodies on factors, look at these activities on transum.

Extension

Many of the more advanced concepts in this lesson are related to prime numbers. They are beyond what you need to know at secondary school but you have already shown that you are capable of understanding some of them. To learn more about perfect number, abundant number, deficient number, amicable number and Goldbach's Conjecture, look at this introduction on mathsisfun.

Try this Product Square or this Factor Pairs of 24 puzzle on transum!

nRich

Take a look at these excellent puzzles to do with factors on nRich: Gabriel's Problem and Product Sudoku. You may also be interested in these nRich Factor Challenges.