The teacher asks for the factors of 24. Your Primary school child can recite the six-times table, so they confidently write 6 and 4—but forget 1, 2, 3, 8, 12 and 24. The next worksheet asks when two repeating events will occur together, and they multiply the two periods without checking whether the answer is the earliest common occurrence. Parents searching for factors and multiples Maths tuition in Bukit Timah are often discovering that knowing multiplication facts and understanding number structure are not quite the same thing.
The core aim of Bukit Timah Mathematics tuition for factors, multiples, prime numbers, HCF and LCM is to help students recognise divisibility, organise factor pairs and explain how numbers relate before memorising shortcut procedures. Primary pupils learn to work with factors, multiples and common factors or multiples at suitable levels. Highest common factor (HCF), lowest common multiple (LCM) and systematic prime factorisation also provide an important bridge into Secondary Mathematics, where the relevant G3 syllabus explicitly covers these techniques. Good teaching makes the number relationships visible and selects the complexity appropriate to the child’s actual school work.
Once a child sees factors as the building blocks of a number and multiples as the numbers reached by repeated equal groups, the topic becomes delightfully orderly. It also begins to help with fractions, ratio, divisibility and algebra.
The quick answer: what is the difference between a factor and a multiple?
A factor of a whole number divides it exactly, leaving no remainder. A multiple of a whole number is obtained by multiplying that number by a whole number. The statements are related: if 6 is a factor of 24, then 24 is a multiple of 6.
The factor list of 24 is 1, 2, 3, 4, 6, 8, 12 and 24. Some multiples of 6 are 6, 12, 18, 24, 30 and 36, and the list continues indefinitely. Factors of a given positive whole number are finite; its positive multiples continue without end.
Children often confuse the two words because worksheets switch quickly between them. A strong tutor asks the student to finish a sentence: “Six is a factor of 24 because 24 ÷ 6 = 4 without remainder.” Or, “24 is a multiple of six because 6 × 4 = 24.” Language and calculation reinforce each other.
Factor pairs make completeness easier
To find the factors of 36, form pairs whose product is 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. Reading both sides of the pairs gives 1, 2, 3, 4, 6, 9, 12, 18 and 36. The pair at 6 × 6 appears only once in the final list.
This strategy is more reliable than writing a handful of familiar numbers and hoping none have been missed. The student can test divisibility systematically until the factor pairs begin repeating in reverse order.
A simple check is to confirm that each candidate factor divides 36 exactly. For example, 36 ÷ 9 = 4, so 9 belongs. But 36 ÷ 5 does not give a whole number, so 5 does not belong.
Prime and composite numbers
A prime number is a positive whole number greater than one with exactly two distinct positive factors: one and itself. Examples include 2, 3, 5, 7, 11 and 13. A composite number is greater than one and has more than two positive factors, such as 4, 6, 8, 9 or 12.
The number 1 is neither prime nor composite. It has only one distinct positive factor, namely one. The number 2 is the only even prime, because every other even positive whole number greater than two is divisible by both 1, 2 and itself.
The purpose is not simply to memorise the primes below 100. Students should understand how to test a number and explain why it is prime or composite. That reasoning will make prime factorisation meaningful later.
Worked example 1: identify whether 29 is prime
To decide whether 29 is prime, test possible small divisors. It is not divisible by 2 because it is odd. Its digit sum is 2 + 9 = 11, so it is not divisible by 3. It does not end in zero or five, so it is not divisible by 5. There is no smaller possible prime divisor up to its square root, approximately 5.39, so 29 is prime.
The square-root observation is a useful extension for students ready to reason about factor pairs. Primary pupils can instead verify using small multiplication facts and systematic division. The point is to use evidence, not guess from the size of the number.
Divisibility rules are useful checks, not random spells
A whole number is divisible by 2 if its final digit is even. It is divisible by 5 if it ends in 0 or 5, and by 10 if it ends in 0. It is divisible by 3 if its digit sum is divisible by 3; a similar digit-sum rule applies for 9. These rules follow from place-value properties of our base-ten number system.
Take 4,572. It is even, so divisible by 2. Its digit sum is 4 + 5 + 7 + 2 = 18, which is divisible by 3 and 9. Therefore 4,572 is divisible by both 3 and 9. A tutor can encourage the student to check one case by division, helping them connect the rule with the actual factor relationship.
Common factors: what two numbers share
The factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The common factors are 1, 2, 3 and 6. The greatest of these is six, so the highest common factor of 18 and 24 is 6.
For a Primary learner working on common factors, listing the small factor sets may be clear enough. Introducing the HCF label and prime-factor technique can help bridge to Secondary Mathematics when appropriate, but should not be used as a reason to rush ahead of the child’s syllabus.
Worked example 2: identical gift packs
A teacher has 48 pencils and 60 erasers. She wants to make the greatest possible number of identical packs using all the items, with every pack containing the same whole-number quantities. The number of packs must divide both 48 and 60 exactly. We therefore want their highest common factor, which is 12.
Each of the 12 packs contains 48 ÷ 12 = 4 pencils and 60 ÷ 12 = 5 erasers. Check the total: 12 × 4 = 48 pencils and 12 × 5 = 60 erasers. Nothing is left over.
This is the kind of story that reveals what an HCF is for. It is about finding a greatest shared grouping size or number of equal groups under the conditions stated, not choosing whichever of the two quantities looks larger.
Common multiples: where two repeating patterns meet
Multiples of 6 include 6, 12, 18, 24, 30, 36, 42 and 48. Multiples of 8 include 8, 16, 24, 32, 40 and 48. Common multiples include 24 and 48, and the lowest positive common multiple is 24.
This is the opposite type of problem from forming a maximum number of equal packs. Here we are looking for the earliest positive number that can be reached by both repeated patterns. The word positive matters because zero is a multiple of every whole number, but the conventional LCM is the least positive common multiple.
Worked example 3: two recurring school activities
One activity repeats every 6 days, while another repeats every 8 days. If both occur today and continue on those schedules, after how many days will they first coincide again? We need the first positive number divisible by both six and eight. The LCM is 24, so they coincide again after 24 days.
A common shortcut is to multiply 6 × 8 and answer 48 days. Forty-eight is indeed a common multiple, but not the smallest. Listing the sequences or identifying overlapping factors shows why 24 is earlier.
To make the problem tangible, draw a simple calendar number line and mark multiples of six above it and multiples of eight below it. The first shared marked position after zero is 24.
Prime factorisation: numbers built from primes
Prime factorisation expresses a positive integer greater than one as a product of prime numbers. For example, 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7. A factor tree can reveal the primes: split 84 into 12 × 7, split 12 into 3 × 4, then split four into 2 × 2.
Different correct factor trees lead to the same set of prime factors, apart from their order. This illustrates the fundamental theorem of arithmetic: each whole number greater than one has a unique prime factorisation up to factor order. Students may not need to state the theorem by name, but the reliability of prime building blocks is worth understanding.
Worked example 4: HCF by prime factorisation
Find the HCF of 36 and 60. Prime factorise: 36 = 2² × 3² and 60 = 2² × 3 × 5. Their shared prime factors are two copies of 2 and one copy of 3. Multiplying them gives 2² × 3 = 12.
A learner should understand why we choose only the factors common to both numbers, with no more copies than either number has. Selecting both factors of 3 from 36 would produce 36, which does not divide 60. The mathematical meaning of “common” controls the calculation.
Worked example 5: LCM by prime factorisation
For the same numbers, 36 = 2² × 3² and 60 = 2² × 3 × 5. A multiple of both needs enough copies of every prime to cover either factorisation. Take the highest power of each prime: 2², 3² and 5. Their product is 4 × 9 × 5 = 180.
Check: 180 ÷ 36 = 5 and 180 ÷ 60 = 3, so 180 is a common multiple. Listing smaller common multiples or using the prime-factor logic confirms that none smaller will include all the required prime powers. This is why HCF and LCM use opposite selection principles.
Worked example 6: two unfamiliar larger numbers
Let 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5. To find the HCF, take the smaller power of each common prime: 2² × 3 = 12. To find the LCM, take the largest necessary power from the combined lists: 2³ × 3 × 5 × 7 = 840.
Notice that 84 × 120 = 10,080, while 12 × 840 = 10,080. For two positive integers, the product of their HCF and LCM equals the product of the integers. This identity can serve as a useful check after a solution, though it should not replace the meaning of either quantity.
HCF and LCM are opposites in the way they select factors
When finding HCF from prime powers, keep only primes shared by both numbers and choose the smaller exponent. When finding LCM, include every prime required by either number and choose the larger exponent. A pupil who swaps these rules will often obtain a value that fails the simplest divisibility test.
Ask the child to check a proposed HCF by dividing both numbers by it. Ask them to check a proposed LCM by dividing it by each given number. These checks make the method self-correcting.
How HCF helps simplify fractions
Consider 18/24. The greatest number that divides both 18 and 24 exactly is six. Divide numerator and denominator by six: 18/24 = 3/4. The value of the fraction has not changed because both parts of the ratio are scaled by the same factor.
A student who repeatedly cancels two and then three is not wrong if the steps are valid, but recognising the HCF can be more efficient. The essential concept is equivalence: 18 of 24 equal units represents the same proportion as three of four larger equal units.
How LCM helps add fractions
Add 1/6 + 1/8. The least positive common multiple of six and eight is 24, so express both fractions in twenty-fourths: 1/6 = 4/24 and 1/8 = 3/24. Their sum is 7/24.
The student can use another common denominator such as 48 and still get the same correct result, but 24 keeps the arithmetic simpler. Teaching should explain why equal denominators are needed: the pieces being added must be the same size.
What factors and multiples reveal in ratio problems
Ratios also rely on common factors. The ratio 18:24 simplifies to 3:4 because both quantities are divided by six. When a student learns that simplifying a ratio preserves a multiplicative relationship, HCF stops being an isolated chapter and becomes a useful tool for broader Mathematics.
The fractions, ratio and percentage companion explains why these relationships are more important than remembering a single operation keyword.
A word problem is not a giveaway that you need HCF or LCM
Children may be tempted to memorise that “greatest” always means HCF and “smallest” always means LCM. Those words are clues, not guarantees. The tutor should ask what is being grouped, what must divide exactly and what quantity needs to occur together.
For identical packs using all supplies, a number of packs must divide every supply count. For repeating activities coinciding, the desired day count must be divisible by each repeat interval. Once that relationship is clear, the HCF or LCM choice is justified.
Where the Primary syllabus ends and Secondary extensions begin
The MOE Primary Mathematics syllabus includes factors, multiples and common factors or multiples at appropriate levels, with variation for Standard and Foundation pathways. More systematic prime-factor methods for HCF, LCM and roots are explicitly part of the 2027 G3 Mathematics K310 number-and-algebra scope.
Parents should not treat every prime-factor technique in this article as a required Primary examination method. Primary pupils first need secure multiplication, division and number relationships. The formal HCF/LCM approach is a useful bridge or extension when suited to readiness and the actual school curriculum.
Six common mistakes worth diagnosing
- Incomplete factor lists: forgetting factor pairs such as 3 and 8 for 24.
- Confusing factors with multiples: writing 30 as a factor of six rather than a multiple.
- Misclassifying one: calling the number 1 a prime number.
- Choosing an arbitrary common multiple: using the product of two periods without checking for an earlier shared multiple.
- Reversing HCF and LCM prime-power rules: picking high exponents for a greatest divisor or low exponents for a common multiple.
- Ignoring problem meaning: deciding the method solely from the word “greatest” or “smallest”.
Each wrong answer provides useful evidence. One student may have insecure times tables; another understands division but forgets the factor list; another is ready for a conceptual explanation of prime powers. A tutor should choose the correction accordingly.
A four-week factors-to-HCF learning plan
- Week 1: make complete factor-pair lists, review multiplication and distinguish factors from multiples.
- Week 2: introduce prime and composite numbers, divisibility checks and common factors or multiples.
- Week 3: practise grouping and repeating-pattern stories, using HCF or LCM language where suitable.
- Week 4: connect the topic to fraction simplification and common denominators; extend into prime factorisation at the appropriate level.
This is an illustrative sequence, not a promise that every child will master HCF and LCM in four weeks. A learner with weak multiplication facts may need more foundation work, while a Secondary student may be ready for more complex prime factorisation and roots.
How 3-pax Maths tuition can make the number structure visible
The immutable eduKateSG three-student Mathematics tutorial reference describes weekly 1.5-hour lessons near Sixth Avenue MRT and close checking of individual workings. A tutor can see whether one child has listed only some factor pairs, another has confused the meaning of a multiple and a third is ready to connect prime factors to algebra.
The value of a small group comes from adapting the next question, not from giving every child the same extra worksheet. A learner should eventually be able to justify which factor or multiple is needed without a hint.
Frequently asked questions
What is the difference between HCF and LCM?
HCF is the greatest positive integer that divides each of the given integers exactly. LCM is the smallest positive integer that is a multiple of each of them.
Is one a prime number?
No. A prime has exactly two distinct positive factors, one and itself. The number one has only one positive factor.
When should a Primary pupil learn prime factorisation?
First ensure multiplication, division, factor pairs and common factors are secure. More advanced prime-factor techniques can serve as suitable extension or as a bridge to Secondary Mathematics, according to the student’s actual syllabus.
How does HCF help with fractions?
Dividing numerator and denominator by their HCF produces the equivalent fraction in simplest form. For 18/24, the HCF is six, giving 3/4.
How do I know whether a word problem needs HCF or LCM?
Ask what the result must do. If the answer must divide several quantities exactly and be as large as possible, HCF may be appropriate. If it must be a common positive multiple and as small as possible, LCM may fit. The story’s conditions must support the choice.
The core aim is recognising how whole numbers fit together
Factors and multiples help children see structure in arithmetic. They explain why a fraction simplifies, why two schedules meet again and how equal packs can be made without leftovers. When students understand the relationships, HCF and LCM are no longer mysterious abbreviations—they are natural answers to well-understood questions.
Continue with Primary number sense and operations, fractions, ratio and percentage and Bukit Timah Mathematics tuition. To discuss persistent multiplication, factors or fraction difficulties, contact eduKate Singapore with a few original school examples.