Mathematics Tutor Singapore | Primary 4 Maths tuition can reveal a surprising confusion. A child knows the four-times table but cannot tell whether four is a factor of twenty-four or twenty-four is a multiple of four. The multiplication fact is familiar; the direction of the relationship is not.
Did you know that one arrangement of 24 counters—six equal groups of four—contains both statements? Four is a factor of twenty-four because it divides the whole exactly. Twenty-four is a multiple of four because it can be made from repeated groups of four. Once those two descriptions are connected, common factors, common multiples and divisibility questions are less mysterious.
Here is a Singapore Primary 4 Mathematics teaching guide covering factors, multiples, equal groups, relevant number patterns and two original word problems. It follows the MOE Primary Mathematics syllabus updated in October 2025 and distinguishes syllabus-aligned work from optional extension, with detailed parent advice rather than promised exam grades.
Open the Primary 4 factors and multiples chapters
- Factors versus multiples: one relationship, two directions
- Worked example: factor pairs of 36
- Find common factors of 24 and 36
- Find common multiples of 4 and 6
- Remainders tell us when something is a factor
- Useful divisibility checks and their limits
- Prime, composite and the special number one
- Worked word problem: equal-sized packets
- Worked word problem: two repeating events
- Greatest and least as extensions, not shortcuts
- Seven wrong ideas I would diagnose first
- Fourteen questions with checked answers
- An achievable Primary 4 learning plan
- Questions parents and pupils ask
- Official syllabus and next Mathematics route
Factors versus multiples: one relationship, two directions
A whole number a is a factor of another whole number b when b can be divided by a with no remainder, with a nonzero. If 24 ÷ 4 = 6 exactly, four is a factor of twenty-four. Reversing the language, twenty-four is a multiple of four because 4 × 6 = 24.
This is a relationship, not two disconnected vocabularies. The product 4 × 6 = 24 immediately shows that four and six are factors of twenty-four, and that twenty-four is a multiple of both four and six. I ask students to say all those facts aloud from one array rather than fill separate factor and multiple worksheets.
| Calculation | Factor statement | Multiple statement |
|---|---|---|
| 4×6=24 | 4 and 6 are factors of 24 | 24 is a multiple of 4 and 6 |
| 3×7=21 | 3 and 7 are factors of 21 | 21 is a multiple of 3 and 7 |
| 5×8=40 | 5 and 8 are factors of 40 | 40 is a multiple of 5 and 8 |
The distinction becomes clear in a counterexample. Twenty-four is a multiple of four, but four is not a multiple of twenty-four within the usual positive whole-number multiplication-table sense. Swapping the labels changes the mathematical claim.
- Ask which number divides the other exactly.
- Draw an equal-group array if the statement seems abstract.
- For a factor, think of an exact divisor of a fixed number.
- For a multiple, think of a result obtained by repeated multiplication of a fixed number.
- Use both directions to check that a proposed relationship is sensible.
Worked example: factor pairs of 36
Imagine arranging 36 square counters into rectangles with complete rows and no spare counters. One row of 36 works; two rows of 18 work; three rows of 12, four of nine and six of six also work. Each rectangle supplies a factor pair.
| Rows | Counters in each row | Factor pair |
|---|---|---|
| 1 | 36 | 1 and 36 |
| 2 | 18 | 2 and 18 |
| 3 | 12 | 3 and 12 |
| 4 | 9 | 4 and 9 |
| 6 | 6 | 6 and 6 |
The full list of positive factors is 1, 2, 3, 4, 6, 9, 12, 18 and 36. Notice that six and six form one pair, not two separate new factor values. Once the row count passes six, the previously listed pairs simply reverse.
A child may keep searching indefinitely. I ask them why stopping at six works. When two positive factors multiply to 36, at least one must be at most √36=6. If both were greater than six, their product would exceed 36. This square-root argument is enrichment; an ordinary multiplication-table explanation is enough to begin with.
Find common factors of 24 and 36
Common factors divide both given numbers exactly. The positive factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The positive factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. The common factors are therefore 1, 2, 3, 4, 6 and 12.
I prefer to teach an organised list first. Each candidate must divide both numbers with no remainder. The number eight divides 24 but not 36, so it cannot appear in the common list. This is a simple mathematical AND condition.
| Candidate | 24 divisible? | 36 divisible? | Common factor? |
|---|---|---|---|
| 3 | Yes | Yes | Yes |
| 4 | Yes | Yes | Yes |
| 6 | Yes | Yes | Yes |
| 8 | Yes | No | No |
| 12 | Yes | Yes | Yes |
A learner who reports all the factors of 24 has not yet interpreted the word common. Another may select only the biggest common factor when asked for all common factors. The last line of the question determines whether a complete list or one particular value is required.
Find common multiples of 4 and 6
A multiple of four appears when four is multiplied by a whole number: 4, 8, 12, 16, 20, 24 and so on. Multiples of six include 6, 12, 18, 24, 30, 36 and so on. The shared results begin 12, 24, 36 and 48.
| Multiples of 4 | Multiples of 6 | Shared values |
|---|---|---|
| 4, 8, 12, 16, 20, 24, 28, 32, 36 | 6, 12, 18, 24, 30, 36 | 12, 24, 36 |
Why is twelve shared? Because 4×3=12 and 6×2=12. A learner can imagine two repeating actions, one every four counts and one every six counts, meeting at the same count. The number twelve is their first positive meeting point.
Common multiples continue indefinitely, so a request to “list all common multiples” without a stated range cannot be satisfied by a finite list. A school question usually specifies the first few, those below a limit or the least positive common multiple.
Remainders tell us when something is a factor
Consider 25 divided by four. We obtain six groups of four with one left over. Because the remainder is one, four is not a factor of twenty-five. But 24 ÷ 4 = 6 with zero remainder, so four is a factor of twenty-four.
A student who writes “four is a factor of 25 because four goes into 25 six times” has overlooked the leftover object. Exact divisibility is the defining condition. A quick counter array can make the one spare item visible.
| Number | Divide by | Quotient and remainder | Factor? |
|---|---|---|---|
| 24 | 4 | 6 remainder 0 | Yes |
| 25 | 4 | 6 remainder 1 | No |
| 35 | 5 | 7 remainder 0 | Yes |
| 37 | 5 | 7 remainder 2 | No |
Useful divisibility checks and their limits
A few patterns can save time. A whole number ending in an even digit is divisible by two. A whole number ending in zero or five is divisible by five. Ending in zero gives divisibility by ten. A number is divisible by three if its digits add to a multiple of three.
For example 234 is divisible by three because 2+3+4=9 and nine is a multiple of three. In fact 234÷3=78. The trick is helpful, but I want the learner to verify it by an actual division on a small example so it does not become magic.
| Test | Helpful check | Example |
|---|---|---|
| Factor 2 | Last digit is even | 246 is divisible by 2 |
| Factor 5 | Last digit 0 or 5 | 245 is divisible by 5 |
| Factor 10 | Last digit 0 | 240 is divisible by 10 |
| Factor 3 | Digit sum divisible by 3 | 234: sum 9, yes |
Not every one-digit divisor has the same last-digit shortcut. We should not teach students to guess unfamiliar divisibility rules without checking. For the P4 task of determining whether a one-digit number divides a given number within 100, organised multiplication facts or division may be entirely appropriate.
Prime, composite and the special number one
As enrichment, a prime number is a whole number greater than one with exactly two positive factors: one and itself. A composite whole number greater than one has more than two positive factors. The number seven is prime; twelve is composite.
What about one? It has only one positive factor, namely itself, so it is neither prime nor composite. This exception is worth explaining instead of treating it as another isolated fact. It helps clarify why “exactly two factors” is part of the prime definition.
Prime and composite vocabulary is useful in later number theory and factorisation. Still, parents should distinguish interesting enrichment from the precise syllabus list: the Primary 4 MOE strand here focuses on factors, multiples, common factors and common multiples with specified small-number constraints.
Worked word problem: equal-sized packets
A school club has 24 blue stickers and 36 red stickers. The organiser wants to make identical packets with no stickers left over, putting the same number of blue stickers in each packet and the same number of red stickers in each packet. What numbers of packets are possible?
The packet count must divide both 24 and 36 exactly. Therefore any positive common factor is possible: 1, 2, 3, 4, 6 or 12 packets. If the organiser specifically wants the greatest possible number of identical packets, the answer is 12.
With twelve packets, each contains 24÷12=2 blue stickers and 36÷12=3 red stickers. Check: twelve packets reproduce the original 24 blue and 36 red without leftovers. The number of packets is a common factor; the contents per packet are quotients.
I would ask a changed question: could we make eight identical packets? No, because eight divides 24 exactly but not 36. The blue count alone is not enough. Students must check every category in the story.
Worked word problem: two repeating events
A fictional toy blinks a green light every four seconds and a blue light every six seconds. Both flash together at the starting time. When do they first flash together again after the start, assuming both continue on their precise schedules?
The green flash times after the start include 4, 8, 12, 16, 20 and 24 seconds. The blue times include 6, 12, 18, 24 and 30. The first positive common value is 12 seconds.
Notice the difference between asking when the lights flash together and asking how many times they have flashed by a certain time. The first uses common multiples. The second requires counting events, including or excluding the starting flash depending on the wording.
Greatest and least as extensions, not shortcuts
The greatest common factor of 24 and 36 is 12. The least positive common multiple of four and six is 12. These terms abbreviate “the largest value in a common-factor list” and “the first positive value in a common-multiple list”.
I would introduce the labels after students understand the lists. Teaching just an algorithm for highest common factor or least common multiple can hide which kind of object the answer represents. The number 12 happened to appear in both of these different examples but not for the same reason.
The mathematical contexts decide which is relevant: splitting two quantities into the greatest number of identical packets uses a common factor, while asking when two repeating schedules meet uses a common multiple.
Seven wrong ideas I would diagnose first
| What the pupil says | Underlying confusion | Next prompt |
|---|---|---|
| 24 is a factor of 4 | Direction of relation reversed | Can 4 be divided by 24 exactly? |
| 8 is a factor of 36 because it is smaller | Size confused with divisibility | Compute 36÷8 and inspect remainder |
| 8 is a common factor of 24 and 36 | Only one number tested | Check both divisions |
| All common multiples are 12,24,36 | Infinite continuation overlooked | What follows after 36? |
| One is prime | Prime definition incomplete | How many distinct positive factors does 1 have? |
| Packet problem solved by listing multiples | Factor versus multiple chosen incorrectly | Is packet count dividing the fixed totals? |
| Two lights meet at 10 seconds | Schedules not listed | Which 4- and 6-second multiples equal ten? |
An important teaching decision is whether the pupil needs better multiplication facts or better interpretation of what those facts mean. Some students have fluent times tables but still confuse an item’s factor with its multiple. They need relationship-building tasks rather than more speed tests.
Fourteen questions with fully checked answers
1. A factor test
Is 7 a factor of 42? Yes, because 42÷7=6 with no remainder.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
2. A non-factor
Is 6 a factor of 35? No; 35÷6 leaves remainder five.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
3. A multiple
Is 48 a multiple of 8? Yes, because 8×6=48.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
4. List factor pairs
Factor pairs of 20 are (1,20), (2,10) and (4,5).
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
5. List all factors
Positive factors of 18 are 1,2,3,6,9,18.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
6. Common factors
Common factors of 12 and 18 are 1,2,3,6.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
7. Common factors again
Common factors of 24 and 36 are 1,2,3,4,6,12.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
8. Multiples of four
First five positive multiples of four are 4,8,12,16,20.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
9. Common multiples
First three positive common multiples of 4 and 6 are 12,24,36.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
10. Compare factor and multiple
For 6×7=42, six is a factor of 42 and 42 is a multiple of six.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
11. Greatest packets
For 24 blue and 36 red stickers with identical packets, 12 is the greatest common packet count, giving 2 blue and 3 red in each.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
12. Repeating alarms
Alarms every 3 and 5 minutes first ring together again after 15 minutes.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
13. Prime extension
Is 11 prime? Yes, its only positive factors are 1 and 11.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
14. The number one
One is neither prime nor composite; it has only one positive factor.
Try a second method: multiplication facts, equal groups or division with remainder. A correct answer should be explainable without relying on a chart supplied by the teacher.
An achievable Primary 4 learning plan
| Session | Teaching focus | Evidence of independence |
|---|---|---|
| 1 | Understand factor vs multiple | Can read one fact in both directions |
| 2 | Make arrays for 24 or 36 | Finds pairs systematically |
| 3 | Divide exactly or with remainder | Tests factor claim |
| 4 | List common factors | Checks both numbers |
| 5 | List common multiples | Extends sequences correctly |
| 6 | Choose between packet and alarm stories | Matches factor or multiple to context |
| 7 | Try efficient divisibility checks | Explains why check works |
| 8 | Compare prime and composite as enrichment | Treats one correctly |
| 9 | Mix question types without headings | Chooses relation |
| 10 | Delayed retest with new numbers | Reasoning survives without hints |
A small group can be useful when children explain why another pupil’s proposed multiple is not a factor. But group size alone is not proof of teaching quality. I would want to see independent explanations and changed-context success, not only an increasing worksheet count.
Questions parents and pupils ask
What is the difference between a factor and a multiple?
A factor divides a given number exactly. A multiple is a product of the given number and a whole number.
Can one number be both a factor and a multiple of another?
Yes, but the direction matters. A positive number is a factor and a multiple of itself; for distinct numbers the claims must be tested separately.
Does a factor always have to be smaller?
A positive factor can equal the number itself. For positive whole-number factors, it cannot exceed the number being factored.
How do I find common factors?
List or test factors of each number and keep values that divide both exactly.
How do I find common multiples?
List multiples of each number and identify shared values, or reason from their repeating schedules.
Is one a prime number?
No. A prime has exactly two distinct positive factors; one has only one.
Are HCF and LCM always the best Primary 4 shortcuts?
Not necessarily. Establish what common factors and multiples represent first and match methods to the official grade scope.
Should every P4 pupil attend Maths tuition?
No. Extra help is useful when a specific persistent gap remains and targeted teaching would help.
Will factors and multiples guarantee PSLE performance?
No. They build number sense, but later achievement depends on a much wider range of mathematical learning.
Official syllabus and next Mathematics route
The MOE Primary Mathematics syllabus updated October 2025 identifies Primary Four factors, multiples, their relationship, testing one-digit factors within 100, common factors and common multiples of two given one-digit numbers. The extra prime discussion and broader number choices above are explanatory extensions, not claims that every such variation is compulsory in Primary Four.
The Mathematics Learning Hub maps the Primary-to-JC journey. For foundational arrays, read Primary 2 Multiplication and Division Word Problems. For number-value confidence, see P1–P4 Number Sense and Place Value. For an extension about factors, explore Why One Is Not a Prime Number.
For a more personalised discussion of the child’s error, arrange an eduKate Singapore parent–student consultation. Bring a basic multiplication question, a factor problem and a common-multiple word problem. Comparing them helps identify whether the bottleneck is number-fact fluency, divisibility or interpreting a situation.