Factors, primes, HCF and LCM
Primes
A prime number is any whole number that can only be divided by itself or 1. Confusingly:
- for the purposes of GCSE and A-level, primes are always positive
- At degree level in Number Theory, the idea gets expanded and they can be negative as well.
Here are the first 10 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
prime number facts:
- 2 is the *only* even prime number
- primes are considered the basic building blocks of all numbers. Nobody has ever worked out a
formula to generate primes, and it is thought currently that this is fundamentally impossible.
There are formulas which generate primes, but only some of them. The only way to generate a list of
all primes up to a limit is to laborious division ( An algorithm called the Sieve of Eratosthenes. )
As such, you just have to learn them. You will need to learn the list above for GCSE.
Compound numbers
All non-prime whole numbers are called compound numbers, and they are made by multiplying two or more primes together.
Examples: 3 x 5 = 15 , 2 x 2 x 7 = 28
Coprime numbers
Two numbers are called 'coprime' if their only common divisor is 1. They *no not* necessarily have to be prime,
although they often are. Two prime numbers are *always* coprime.
Examples:
3 and 5 : Both prime, therefore they are also coprime ( 1 is the only number that will divide into both of them )
8 and 9 : Both compound numbers, but 1 is still the only number that can divide into both of them, therefore coprime.
Fact:
Technically, this is how you tell a fraction is in lowest form: The numerator and denominator are coprime.
Factors
A factor is any number that divides exactly into another number without a remainder. It's basically times tables in reverse.
Examples:
4 is a factor of 28, because 28 ÷ 4 = 7 , a whole number.
3 is a factor of 27, because 27 ÷ 3 = 9 , a whole number.
Some numbers have lots of factors, some not very may at all.
30, 36, 60 and 72 are lowish numbers that famously have lots of factors, so they come up a lot in factor questions.
For example, the factors of 36 are 1 , 2 , 3 , 4 , 6, 9, 12 and 18.
Factor pairs
Some numbers (like 36) can be made conveniently by just two factors:
36 = 1 x 36 , 2 x 18 , 3 x 12 , 4 x 9 , 6 x 6 .
Multiples
..are the numbers in times-table. So for example, for 10, its multiples are 20, 30, 40, 50 and so on, forever.
When a question says "identify a multiple of 8 from the list" it just means a number in the 8-times-table.
HCF - the Highest Common Factor
When two numbers are *not* coprime then this means there is a number or numbers other than 1 and themselves that will
divide into them. Sometimes there is only one common factor, sometimes there are many. The HCF refers to the *biggest*
Number that will do this.
Example: Two numbers 20 and 36. They can both be divided by 2, so this is a 'common factor.' But it is not the
*highest* common factor, which is actually 4.
LCM - lowest common multiple
When you write out the times tables ( to infinity! ) for two numbers, there will always be an infinite number
of matching numbers that exist in both lists. The LCM is the *lowest* match.
Example: 3 and 5:
3: 3 6 9 12 15 18 21 24 27 30
5: 5 10 15 20 25 30
- You can see that 30 exists in both lists, so it is a 'common multiple.' But it is not the lowest - this is 15.
Fact:
LCM used to be called 'lowest common denominator' back in the day, but the problem was that really referred to LCM's in
addition calculations with fractions, so they changed it to the more generic LCM.
Finding the HCF and LCM of two numbers
In GCSE maths, you have to learn a really long-winded way of doing this:
- draw the prime factor trees for both numbers
- draw a venn diagram and correctly fill it with the numbers in the trees
- the HCF is the Venn middle intersection numbers multiplied together
- the LCM is all numbers in the Venn multiplied together
There is really no need to do this. The easiest way is to just write the numbers as a ratio, and simplify.
Example: Find the HCF and LCM of 30 and 54.
30 : 54 They are both even, so divide by 2:
15 : 27 They both divide by 3, so do this to give:
5 : 9
The numbers are now 'coprime' so they are in simplest form.
- we divided by 2 and then 3, so just multiply these together: 2 x 3 = the HCF.
- to get the LCM, just cross-multiply the bottom numbers with the top; either way will work so you
just choose what looks the easiest:
9 x 30 = 270 or you can do 5 x 54 = 270 ( they always come to the same answer. ) 270 is your LCM.
Additional facts
- You are rarely told this at school but it is a fact that two numbers a and b, when multiplied together,
make the same result as thier HCF and LCM multiplied together.
Using the example above of 30 and 54, 30 x 54 = 1620, and 6 x 270 is also 1620.
This could *potentially* come up as a GCSE question:
"The HCF and LCM of two numbers a and b are 6 and 270. If 'a' is 30, what is 'b' ?
To do this, you do ( HCF x LCM ) ÷' b , so ( 6 x 270 ) ÷ 30. This will give 54.
- When dealing with absolutely massive primes, it is a general principle of number theory that if you start
with two enormous prime numbers, and multiply them together to get an even more massive compound number,
it is extremely difficult to work out the original primes without using large banks of supercomputers.
Virtually the entire security of the internet rests on this principle. When you connect to your bank or
Ebay or Amazon to make a secure purchase ( secure in that nobody tapping your internet line an tell what
your passwords are, ) you are relying on this principle, as the 'secret keys' used for internet transations
Are generated by multiplying two massive primes together.
Difficulty Levels
The same questions are tested at levels 1 2 and 3 but with bigger numbers.
GCSE will be about levels 1 to 2.