What you'll learn
A factor of a number is a whole number that divides into it exactly, with nothing left over; a multiple is what you get when you multiply a number by a whole number. So 6 is a factor of 24 because 24 ÷ 6 = 4 exactly, and 24 is a multiple of 6 because 6 × 4 = 24. A prime number has exactly two factors, itself and 1, and a square number is what you get when a whole number is multiplied by itself, like 7 × 7 = 49. This topic covers listing factors in pairs, finding common factors and common multiples, recognising primes up to 50, knowing the square numbers up to 144, and using quick divisibility tests. These ideas underpin fractions, ratio and problem-solving right across the paper, and Number is 19 of the 40 items on the SEA Mathematics paper.
Key terms and definitions
Factor — a whole number that divides exactly into another number. The factors of 10 are 1, 2, 5 and 10.
Factor pair — two factors that multiply to give the number, such as 3 and 8 for 24.
Multiple — the result of multiplying a number by a whole number. The multiples of 5 are 5, 10, 15, 20 and so on.
Common factor — a factor shared by two or more numbers. 4 is a common factor of 12 and 20.
Highest common factor (HCF) — the largest factor that two numbers share.
Common multiple — a multiple shared by two or more numbers.
Lowest common multiple (LCM) — the smallest multiple that two numbers share.
Prime number — a number with exactly two different factors, 1 and itself. 2, 3, 5, 7 and 11 are prime.
Composite number — a number with more than two factors, such as 4, 6, 9 or 100.
Square number — the product of a whole number multiplied by itself: 1, 4, 9, 16, 25 and so on.
Square root — the number that was squared. The square root of 81 is 9, written √81 = 9.
Core concepts
Finding all the factors of a number
The safe method is to hunt in pairs, starting at 1 and working upwards. For 24:
| Try | Pair |
|---|---|
| 1 | 1 × 24 |
| 2 | 2 × 12 |
| 3 | 3 × 8 |
| 4 | 4 × 6 |
| 5 | does not divide |
| 6 | 6 × 4 — already have it, so stop |
Reading the pairs out gives the factors in order: 1, 2, 3, 4, 6, 8, 12, 24. That is eight factors.
You stop as soon as a pair repeats itself the other way round. That point always arrives, and it is what makes the method reliable — children who hunt at random almost always miss one.
Every number has at least two factors, 1 and itself, except the number 1, which has only one factor.
Multiples go on forever
The multiples of a number are just its times table, continued. The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 and so on with no end. A number has a limited number of factors but an unlimited number of multiples, and mixing the two up is the biggest single trap in this topic.
A useful sentence to say aloud: factors are small and few, multiples are big and many. Factors of 12 stop at 12; multiples of 12 start at 12.
Common factors and the HCF
To find what two numbers share, list both sets and look for the overlap.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
The common factors are 1, 2, 3, 4, 6 and 12, so the highest common factor is 12.
This has a real use. If Riya has 24 mangoes and 36 oranges and wants to make identical gift bags with none left over, the greatest number of bags she can make is 12 — each holding 2 mangoes and 3 oranges.
Common multiples and the LCM
List the multiples of each number until one appears in both lists.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28 Multiples of 6: 6, 12, 18, 24, 30
The common multiples are 12, 24 and so on, and the lowest common multiple is 12.
LCM questions in SEA nearly always look like this: two things happening at different regular intervals, and the question asks when they next happen together.
Prime numbers
A prime number has exactly two factors. Nothing else divides into it.
The primes below 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Two facts are worth memorising because they are asked directly:
- 1 is not prime. It has only one factor, and a prime must have exactly two.
- 2 is prime, and it is the only even prime. Every other even number has 2 as an extra factor, so it has at least three.
Any number that is not prime (and is not 1) is composite. 9 is composite because 3 divides it; 51 is composite because 3 × 17 = 51, which surprises people because 51 looks prime.
Square numbers
A square number is a whole number times itself. They are called squares because that many counters can be arranged in a perfect square: 16 counters make a 4-by-4 square.
| Number | Square |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
Knowing this list by heart is worth real marks, because square numbers turn up in area questions as well as number questions.
Square numbers have one odd property: they have an odd number of factors. 36 has nine factors (1, 2, 3, 4, 6, 9, 12, 18, 36) because the pair 6 × 6 contributes only one new factor instead of two. Every non-square number has an even number of factors.
Divisibility tests
These save a great deal of time and prevent guessing.
| Divisible by | Test | Example |
|---|---|---|
| 2 | last digit is 0, 2, 4, 6 or 8 | 418 ✔ |
| 3 | digits add to a multiple of 3 | 342 → 3+4+2 = 9 ✔ |
| 4 | last two digits divide by 4 | 3,516 → 16 ✔ |
| 5 | last digit is 0 or 5 | 675 ✔ |
| 6 | passes the tests for both 2 and 3 | 342 ✔ |
| 9 | digits add to a multiple of 9 | 342 → 9 ✔ |
| 10 | last digit is 0 | 4,290 ✔ |
Checking 342: it is even, and its digits add to 9, so it divides by 2, 3, 6 and 9. Indeed 342 ÷ 9 = 38 and 342 ÷ 6 = 57.
Worked examples
Example 1: Listing every factor without missing one
Write down all the factors of 48.
Work through in pairs, starting at 1:
1 × 48, 2 × 24, 3 × 16, 4 × 12. Five does not divide 48 and neither does 7. Then 6 × 8. The next number to try is 7, already ruled out, and 8 would give 8 × 6, which repeats — so stop.
Reading the pairs from the outside in gives:
1, 2, 3, 4, 6, 8, 12, 16, 24, 48 — ten factors.
Because 48 is not a square number, it has an even number of factors, which is a quick sense-check that none has been dropped.
Example 2: Two buses and the lowest common multiple
At the terminus, the Arima bus leaves every 12 minutes and the Sangre Grande bus leaves every 18 minutes. Both leave together at 8:00 a.m. At what time do they next leave together?
List the multiples until they meet.
Multiples of 12: 12, 24, 36, 48 Multiples of 18: 18, 36, 54
The lowest common multiple is 36, so the buses next leave together 36 minutes after 8:00 a.m., which is 8:36 a.m.
Notice that 12 × 18 = 216 is also a common multiple, but it is not the lowest one, and the question asked for the next time.
Example 3: A number riddle
Devon is thinking of a number. It is a square number, it is between 20 and 50, and it is an even number. What is his number?
Take one clue at a time.
The square numbers between 20 and 50 are 25 (5 × 5), 36 (6 × 6) and 49 (7 × 7).
Of those, only 36 is even, because 25 and 49 both end in an odd digit.
Devon's number is 36.
Common mistakes and how to avoid them
Muddling factors with multiples. Writing "the factors of 6 are 6, 12, 18". Fix: repeat the sentence "factors are small and few, multiples are big and many", and remember that a number's largest factor is itself.
Missing a factor when listing. Fix: always hunt in pairs from 1 upwards and stop only when a pair repeats. Never list at random.
Calling 1 a prime number. Fix: a prime needs exactly two factors, and 1 has one.
Saying 2 is not prime because it is even. Fix: 2 has only the factors 1 and 2, so it qualifies. It is the only even prime.
Assuming every odd number is prime. 9, 15, 21, 25, 27, 33, 35, 39, 45, 49 and 51 are all odd and all composite. Fix: run the divisibility tests for 3, 5 and 7 before declaring a number prime.
Confusing a square number with double the number. Writing 6² as 12. Fix: read it aloud as "six times six", never "six twos".
Giving a common multiple instead of the lowest one. Fix: underline the word "lowest" or "next" in the question.
How parents can help at home
None of this needs you to remember school mathematics — it needs you to ask the right question.
Play "factor or multiple?" Say a pair of numbers, such as 4 and 20, and ask which is the factor and which the multiple. Thirty seconds at a time, often, beats a long session.
Use packing and sharing at home. "We have 30 sweets and I want equal bags with none left over — how many bags could we make?" That is factor-hunting in disguise, and the answer set (1, 2, 3, 5, 6, 10, 15, 30) comes straight from the pairs method.
Point at door numbers and car plates. "Is 57 prime?" It is not — 3 × 19 = 57 — but the guessing and checking is the whole exercise.
When your child is stuck listing factors, say "do it in pairs, start at one". That single prompt fixes most of the errors in this topic without you needing to know the answer yourself.
Test the squares like times tables. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Ask them backwards too: "what times itself makes 64?"
Exam technique for factors, multiples, primes and square numbers
The SEA Mathematics paper is weighted 100 marks against 60 for ELA and 40 for ELA Writing, and it has 40 items worth 75 marks. Number contributes 19 of those items, so questions of this kind appear every year and are usually quick marks.
Read the key word carefully. "Factor", "multiple", "prime", "square", "common" and "lowest" each change the answer completely, and the wrong multiple-choice options are built precisely from swapping one of them.
Use the divisibility tests rather than long division whenever you are testing whether a number is prime. For a number under 100 you only need to try 2, 3, 5 and 7.
When a question asks for "the greatest number of equal groups", think HCF. When it asks "when do they next happen together", think LCM. Those two phrasings cover almost every problem-solving item on this topic.
Write out short lists in full. Trying to hold the factors of 48 in your head is how one gets lost; ten numbers on the working line take fifteen seconds and can be checked.
Quick revision summary
- A factor divides into a number exactly; a multiple is the number multiplied up.
- List factors in pairs from 1 and stop when a pair repeats.
- Factors are few and stop at the number itself; multiples are unlimited.
- The HCF is the largest shared factor; the LCM is the smallest shared multiple.
- A prime has exactly two factors. 1 is not prime; 2 is the only even prime.
- Primes below 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
- Square numbers to 144: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
- Square numbers have an odd number of factors; all other numbers have an even number.
- Divisibility: even for 2; digit sum for 3 and 9; last two digits for 4; ends 0 or 5 for 5.
- "Greatest equal groups" means HCF; "next time together" means LCM.