My students normally arrive well versed in finding prime factors. They enjoy drawing factor trees and generally prefer to call them ‘cherry trees’.

It is worth stressing to students that 1 is not a prime number. However students are often interested that this was not always the case and as recently as the 1950s some mathematicians classed 1 as a prime.

The most common error I see in factor trees is in the very last step where students split 6 into 3 and 3 or 10 into 5 and 5. Sadly this often happens after many correct steps.

What to do after learning how to find prime factors can often be the difficult question and I have seen many students being given larger and larger numbers to decompose using more and more steps. Although time consuming this can be a valuable exercise if time is spent looking at divisibility tests. For example a many-digit number is divisible by 2* ^{n}* if the final

*n*digits are divisible by

*n*.

Generally teachers then move on to Highest Common Factors (HCFs) and Lowest Common Multiples (LCMs). Now that Venn Diagrams are part of the new GCSE this is a good opportunity to consolidate both skills.

For example to find the HCF or LCM of 80 and 72 write their prime factors in a Venn diagram.

The HCF is the product of the factors in the *intersection* of the two sets (2 × 2 × 2) and the LCM is the product of all the numbers in the *union* of the two sets (2 × 2 × 2 × 2 × 3 × 3 × 5).

An activity I like to use is to get my students to write 10, 100 and 1000 as products of prime factors using indices and then to ask them to write a googol (10^{100}) as a product of prime factors using indices. This helps to given them a sense of the size of each number.

You could then discuss the practical uses of a googol for describing quantities. How do the number of hairs on a human head compare with the number of insects in a swarm or the number grains of sand on a beach. Does a googol describe the number of stars in the sky or the number of atoms in the universe?

(The only one that comes close is the number of atoms in the universe at ~10^{80}.)

*Look out for my next post for an in-depth look at how to estimate the number of atoms in the universe.*

Many thanks,

Steve Cavill

*Steve Cavill BSc(Hons) PGCE FCIEA has taught maths in both state and independent schools. He spent a few years as an Associate Lecturer for the OU and has written a number of GCSE maths books, workbooks and revision guides as well as being a senior examiner and moderator for GCSE and IGCSE.*