Why would we care about factors? Because mathematics, unlike human structures like language, has rules that don’t break. In maths, when a rule has an exception the rule must be rewritten.
First problem: do we understand what a factor is? Does everyone agree that they do understand what we mean by a factor?
We’re only working with positive integers here - what back in primary school you called ‘numbers’. None of those pesky decimal things. So when someone asks if 28 divides by seven the answer is ‘yes’, but if 30 can be divided by 7 (same thing, different words), the answer is ‘no’. Meaning, “No, not without having a remainder or some decimal bits”. This is a good argument for knowing your tables, too.
So the point, of doing this is to use stuff we already know (or are supposed to know), like how to recognise an even number (automatically it can be divided by two, so two is a factor). Or, because a number ends in a zero, it must be divisible by 10, so not only is 10 a factor, butr so are the factors of ten, i.e. 2 and 5.
One consistent way to tackle factorisation—at least until the ideas are well understood—is to divide exhaustively by successive primes. The primes are {2,3,5,7,11…} so, for example,
180 = 2x90 = 2x2x45 = 2x2x3x15 = 2x2x3x3x5
144 = 2x72 = 2x2x36 = 2x2x2x18 = 2x2x2x2x9 = 2x2x2x2x3x3
..and one way to write this is 180 = 2².3².5 while 144 = 2⁴.3², known as the canonical form. It is quite convenient and unique, that is, no two numbers can have the same canonical form. I think that entirely obvious and therefore difficult to prove satisfactorily. If you’re converting a number to canonical form, you divide by two until you can’t (i.e. use the last integer result), then by 3, then by 5, then 7 and so on.
Exercise
1. Write these numbers in canonical form 40320, 9216, 121275
2. Write these numbers in canonical form 1234, 4567, 5678, 2468.
3. Write down all the factors of these numbers 2 (two factors), 6 (four factors) , 28 (six factors), 496 (how many do you think there should be?). Use these answers for Q10 on the next page.
4. What sort of number has an odd number of factors?
5. A factorial number such as n! (“en factorial”) is 1.2.3.4…..n These become large very quickly. Write a table of the first 10 factorials, including canonical form. Check 8! against your answer in Q1.
1. 40320=8! = 2⁷.3².5.7 9219 = 9x1024 = 210.3² 121275 = 3².5².7².11
2. 1234= 2x617 4567 is prime, 5678 = 2.17.167, 2468 = 2².617
3. See Q10 in Factors 1 4. Only squares can have an odd number of factors. Primes have exactly two factors (itself and 1). 5.
