Factors and Multiples: How to Tell Them Apart — Common Factors, Common Multiples, HCF and LCM Explained with Examples
A factor is a number that divides a given number exactly; a multiple is what you get when you multiply a number by a whole number. The two point in opposite directions: a factor is not greater than the original number, and a multiple is not less than it. A common factor is a factor shared by two numbers, and the largest of these is the highest common factor (HCF). A common multiple is a multiple shared by two numbers, and the smallest of these is the lowest common multiple (LCM). To find the HCF you can list the factors or use short division; to find the LCM you can list the multiples or use short division. Take 12 and 18: the common factors are 1, 2, 3 and 6, so the HCF is 6; the first few common multiples are 36, 72 and 108, so the LCM is 36. A prime number is a number with only two factors, 1 and itself, and is an important foundation for understanding factors.
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- A factor is not greater than the original number and a multiple is not less than it; use size to judge the direction.
- A common factor is a shared factor, and the largest one is the HCF.
- A common multiple is a shared multiple, and the smallest one is the LCM.
- The HCF of 12 and 18 is 6, and the LCM is 36.
- A prime number has only two factors, 1 and itself, and is the basis of prime factorisation.
- In word problems, first decide whether it is 'at most' or 'at least', then choose the method.
First, get the direction of factors and multiples right
Factors and multiples describe the same multiplication-and-division relationship; you are just looking at it from different directions. If 3 times 4 equals 12, then 3 and 4 are both factors of 12, and 12 is a multiple of 3 and also of 4. Put simply, a factor is 'a number that divides this number exactly', and a multiple is 'what you get when you multiply this number by a whole number'.
You can use size to judge the direction: the factors of a number are never greater than the number itself, and the multiples of a number are never less than the number itself (not counting 0 times). For example, the largest factor of 12 is 12, while the multiples of 12 can go on for ever: 12, 24, 36… When pupils are confused, ask them first 'is this number bigger or smaller than the original number?', then decide whether it is a multiple or a factor.
Another common misunderstanding is treating 0 as a factor. 0 cannot be a factor, because in division the divisor cannot be 0; but 0 is a multiple of any non-zero whole number, though at primary level this is usually only mentioned as an aside.
A prime number is a whole number greater than 1 with only two factors, 1 and itself. For example, 2, 3, 5, 7 and 11 are all prime numbers; their only factors are 1 and themselves. Knowing about prime numbers helps pupils work out how many factors a number has, because prime numbers have the fewest factors — just two.
Every whole number greater than 1 can be written as a product of prime numbers, a process called prime factorisation. For example, 12 equals 2 times 2 times 3, and 18 equals 2 times 3 times 3. Prime factorisation is an advanced method for finding the HCF and LCM; pupils should first become fluent at listing factors and multiples before tackling it.
Using 12 and 18 to see common factors and the HCF clearly
Common factors are simply the factors that two numbers 'share'. Take 12 and 18: first list the factors of each. The factors of 12 are 1, 2, 3, 4, 6 and 12; the factors of 18 are 1, 2, 3, 6, 9 and 18. The numbers 1, 2, 3 and 6 that appear on both lists are the common factors, and the largest of them, 6, is the highest common factor.
There are two usual ways to find the HCF. The first is to list all the factors and take the ones in common; this suits questions with smaller numbers, and its advantage is that the process is clear. The second is short division: divide both numbers by a common factor until there is no common factor greater than 1, then multiply the divisors on the left. For example, divide 12 and 18 by 2 to get 6 and 9, then divide by 3 to get 2 and 3; the 2 and 3 on the left multiply to 6, which is the HCF.
The HCF is often used in 'sharing equally into groups' problems. For example, if you share 12 pens and 18 notebooks equally into a number of portions, with the same number of pens and notebooks in each portion, the greatest number of portions is the HCF, 6, with 2 pens and 3 notebooks in each portion.
If pupils have already mastered prime factorisation, they can also use it to find the HCF: write each number as a product of primes, then take the prime factors that appear in both and multiply them. For example, 12 equals 2 times 2 times 3 and 18 equals 2 times 3 times 3; the shared part is one 2 and one 3, which multiply to 6. This method is especially convenient with larger numbers.
Using the same pair of numbers to see common multiples and the LCM
Common multiples are simply the multiples that two numbers 'share'. The multiples of 12 are 12, 24, 36, 48, 60, 72…; the multiples of 18 are 18, 36, 54, 72, 90… The numbers 36, 72… that appear on both lists are the common multiples, and the smallest of them, 36, is the lowest common multiple.
The LCM can also be found by short division: multiply together all the divisors on the left and the quotients in the bottom row. Take 12 and 18: after short division the divisors on the left are 2 and 3, and the bottom row is 2 and 3, so the LCM is 2 times 3 times 2 times 3, which equals 36. Another way is to find the HCF first, then use 'the product of the two numbers divided by the HCF'; 12 times 18 divided by 6 also equals 36.
The LCM is often used in 'happening at the same time' problems. For example, if A is on duty every 12 days and B every 18 days, then after a day when both are on duty together, they will next be on duty together after 36 days; this 36 is the LCM. When doing questions, first decide whether the question asks 'into how many portions at most' or 'how long at least must we wait', and you can tell whether to use the HCF or the LCM.
Prime factorisation can also be used to find the LCM: write out the prime factorisation of both numbers, take the highest power of each prime factor, then multiply. For example, 12 is 2 squared times 3, and 18 is 2 times 3 squared; take 2 squared and 3 squared, and 4 times 9 equals 36.
Revision methods that really make the concepts clear
The main reason these four terms are easily confused is that pupils memorise the definitions without connecting them to concrete examples. When revising, prepare a comparison table with 'factor, multiple, common factor, common multiple' on the left, and on the right an example you have worked out yourself for each, plus one sentence explaining the difference. Writing it out yourself once is more effective than reading the textbook over and over.
When practising, start with small numbers: try 6 and 8 first, then 12 and 18, and only then larger numbers. After each question, ask the pupil to say in one sentence what the answer represents, for example '6 is the HCF of 12 and 18, so they can be split into 6 portions at most'. This is how calculation and meaning get connected.
With word problems, first circle the key words: 'at most' and 'share equally' usually point to the HCF; 'at least', 'at the same time' and 'again' usually point to the LCM. This deciding step is worth practising repeatedly, because marks are often lost not through calculation errors but through choosing the wrong method.
Once pupils know about prime numbers and prime factorisation, encourage them to check for themselves: list the prime numbers from 1 to 30, then pick two numbers at random, do a prime factorisation, and check the answer with short division. This habit of self-checking deepens understanding and reduces reliance on rote memorisation.
The Hong Kong primary mathematics curriculum generally introduces factors, multiples, prime numbers and prime factorisation between Primary 4 and Primary 6, and these concepts are applied to algebraic work in secondary school. Parents can follow the school's teaching schedule and arrange focused revision in the relevant year, without rushing to learn ahead.
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When extra help is needed
Factors and multiples are an important foundation in the upper primary years, and they are still used in secondary school algebra (for example, factorising). If a pupil still cannot tell the two apart in Primary 6, it is best to deal with it early so that later learning is not affected. Parents can first observe whether the pupil's 'definitions are unclear' or their 'calculation is not fluent', as the two need different approaches.
If you are considering extra support, there are various forms of maths tutoring on the market, such as tutoring centres, private tutoring, home tutoring and online tutoring. They differ in arrangements, and fees vary with the format, district and tutor's qualifications, generally within a floating range; parents should find out the actual arrangements before deciding. The Consumer Council has consumer information on various types of services that can be used for reference.
Whether or not there is tutoring, the most effective method is still to have the pupil explain the concepts themselves. Parents can ask the pupil to demonstrate finding the HCF and LCM with 8 and 12 on the spot; from the explanation you can see how well they understand.
If the school curriculum focuses on Primary 5 or Primary 6, parents can first check the key topics for that year to see whether the pupil has already met prime numbers and prime factorisation, then decide the order of revision.
Frequently asked questions
What is the most confusing thing about factors and multiples?
The most confusing thing is direction. A factor is a number that divides the original number exactly and is not greater than it; a multiple is what you get when you multiply the original number by a whole number and is not less than it. For example, 6 is a factor of 12, and 12 is a multiple of 6. Comparing sizes first reduces mistakes.
What are the HCF and LCM of 12 and 18?
The factors of 12 are 1, 2, 3, 4, 6 and 12, and the factors of 18 are 1, 2, 3, 6, 9 and 18; the common factors are 1, 2, 3 and 6, so the HCF is 6. The first few common multiples of 12 and 18 are 36, 72 and 108, so the LCM is 36.
How do you decide whether to use the HCF or the LCM in a word problem?
If the question uses words such as 'into how many portions at most' or 'share equally', it usually uses the HCF; if it uses 'how long at least must we wait' or 'at the same time again', it usually uses the LCM. Before calculating, write in one sentence what the answer represents; this effectively reduces choosing the wrong method.
What is a prime number? How is it related to factors?
A prime number is a whole number greater than 1 with only two factors, 1 and itself, for example 2, 3, 5, 7 and 11. Every whole number greater than 1 can be broken down into a product of prime numbers, called prime factorisation, and this method can be used to find the HCF and LCM.
Should Primary 5 pupils learn factors or multiples first?
They are best learned together, because they are the two directions of the same multiplication-and-division relationship. Primary 5 pupils can start by listing factors and multiples, then gradually move on to common factors, common multiples, prime numbers and prime factorisation. The Hong Kong primary curriculum generally covers these topics in Primary 5 and Primary 6; simply follow the school's schedule.