BTT Mathematics / Primary Mathematics Learning Hub / Guide 9
Factors describe exact groupings inside a number. Multiples describe repeated copies of a number. These two ideas connect times tables, division, prime numbers, divisibility, common factors and common multiples. They are not a bag of unrelated tests.
If twenty-four counters can be arranged as four equal groups of six, then four and six are factors of twenty-four. The same statement can be read forward: twenty-four is a multiple of four and a multiple of six. One arrangement gives both languages.
This guide begins with arrays and factor pairs, then develops divisibility, prime and composite numbers, common factors, highest common factor, common multiples and lowest common multiple. Select sections according to current school scope in the MOE Primary Mathematics syllabus.
Factors · Multiples · Divisibility · Prime and composite · Common factors and multiples · 24 questions · Worked answers
1. Factors are exact group sizes or exact group counts
Arrange twelve counters into one row of twelve, two rows of six or three rows of four. Each arrangement uses all twelve counters with no remainder. Therefore the factor pairs are 1 and 12, 2 and 6, 3 and 4.
A factor divides a whole number exactly. If 12 ÷ 3 = 4 with no remainder, then three is a factor of twelve. If 12 ÷ 5 leaves a remainder, five is not a factor of twelve.
The word exact matters. A divisor can be used in a division even when there is a remainder, but it is a factor only when the division has remainder zero.
Find factors systematically
To find every factor of thirty-six, test possible factor pairs from one upward. 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6 all work. Once the smaller factor has reached six, later pairs would repeat earlier factors in reverse order.
The factors are 1, 2, 3, 4, 6, 9, 12, 18 and 36. Listing them in order makes omissions easier to notice.
One and the number itself are always factors
Every positive whole number has one as a factor and itself as a factor. That follows from 1 × n = n.
This does not mean every number has many factors. Seven has only one and seven. Twelve has six factors. The number of possible equal-group arrangements varies.
Square numbers create a repeated factor pair
Thirty-six has factor pair 6 × 6. The pair does not produce two different factor values; six appears once in the factor list. This is why square numbers have an odd number of positive factors while many non-square numbers have factors paired in distinct twos.
The general statement about factor counts is an extension; primary learners can simply notice that the middle pair of a square repeats the same factor.
2. Multiples are repeated copies
The multiples of six begin 6, 12, 18, 24, 30, 36 and continue without end. Each can be written as six multiplied by a positive whole number.
If twenty-four is a multiple of six, then six is a factor of twenty-four. These are two directions of the same relationship.
A multiple list has no final positive term
Factors of a fixed positive number are limited because only finitely many whole-number groupings fit exactly inside it. Multiples continue because we can always make one more group.
So “list all the multiples of five” is impossible if no upper limit is given. A better instruction is “list the first six positive multiples of five” or “list the multiples of five below forty.”
Use skip counting as structure, not just recital
Counting 7, 14, 21, 28, 35 is useful because each step adds one more group of seven. If a learner loses track, connect the sequence to 1 × 7, 2 × 7, 3 × 7 and so on.
This allows a missing multiple to be reconstructed rather than recalled as a chant.
Multiples can be compared through common landmarks
The multiples of four include 4, 8, 12, 16, 20, 24. The multiples of six include 6, 12, 18, 24. Twelve and twenty-four appear in both lists. They are common multiples.
The smallest positive common multiple is twelve. That value becomes important in repeating schedules and fraction denominators.
3. Divisibility rules compress repeated checking
A divisibility rule gives a quick way to decide whether division will have remainder zero. The rule is useful because it reflects place-value structure; it should not replace the meaning of exact division.
Divisible by 2
A whole number is divisible by two when its ones digit is even: 0, 2, 4, 6 or 8. This works because every complete ten is already divisible by two, so only the ones determine whether an unpaired unit remains.
Thus 3,748 is divisible by two because the final digit is eight. 3,749 is not.
Divisible by 5 and 10
A whole number is divisible by five when its ones digit is zero or five. It is divisible by ten when its ones digit is zero.
Every multiple of ten is also a multiple of five, but not every multiple of five is a multiple of ten. Twenty-five shows the difference.
Divisible by 3 and 9
For divisibility by three, add the digits. If the digit sum is divisible by three, so is the original number. For divisibility by nine, the digit sum must be divisible by nine.
For 4,572, the digit sum is 4 + 5 + 7 + 2 = 18. Eighteen is divisible by both three and nine, so 4,572 is divisible by both.
The rule comes from the fact that powers of ten differ from one by multiples of nine, but a full proof belongs later. At Primary level, use the rule together with actual division checks on examples.
Divisible by 4
A whole number is divisible by four when the number formed by its final two digits is divisible by four. All complete hundreds are divisible by four, so only the final two digits decide the remainder.
Therefore 7,316 is divisible by four because sixteen is divisible by four. 7,318 is not.
Do not combine rules carelessly
A number divisible by two and three is divisible by six. But being divisible by four and six does not automatically mean the least common multiple is twenty-four; twelve is already divisible by both.
Rules identify divisibility relationships. Common-factor and common-multiple questions still require reasoning about the quantities involved.
4. Prime and composite numbers describe factor structure
A prime number greater than one has exactly two positive factors: one and itself. A composite number greater than one has more than two positive factors.
Seven is prime because its only factors are one and seven. Twelve is composite because it has factors 1, 2, 3, 4, 6 and 12.
One is neither prime nor composite
One has only one positive factor. It therefore does not meet the definition of prime, which requires exactly two. It also does not meet the definition of composite, which requires more than two.
This classification is important because prime factorisation and many later results depend on keeping one separate.
Two is the only even prime
Every even number greater than two is divisible by two and therefore has at least the factors one, two and itself. Two itself has only factors one and two, so it is prime.
Test a possible prime efficiently
To decide whether twenty-nine is prime, there is no need to test every smaller number. Try possible small divisors. It is not even, its digit sum is eleven so it is not divisible by three, and it does not end in zero or five. Since the next possible factor pair would involve a factor greater than the square root of twenty-nine, no further small factor is needed.
The square-root shortcut is an extension. The Primary idea is to test factor pairs systematically and stop once the smaller factor would exceed its paired factor.
Prime factorisation as a multiplication skeleton
Thirty-six can be broken into 6 × 6, and each six is 2 × 3. So 36 = 2 × 2 × 3 × 3.
Different factor trees lead to the same prime factors apart from order. This provides a stable description useful for common-factor and common-multiple work.
5. Common factors and common multiples answer different questions
Highest common factor
Find the factors of eighteen: 1, 2, 3, 6, 9, 18. Find the factors of twenty-four: 1, 2, 3, 4, 6, 8, 12, 24. Their common factors are 1, 2, 3 and 6. The highest common factor is 6.
HCF problems often involve making the largest possible equal group size that divides several quantities exactly.
Worked example A: Make identical packs
There are eighteen red pencils and twenty-four blue pencils. We want the greatest number of identical packs, with every pack containing the same number of red pencils and the same number of blue pencils, and no pencils left over.
The number of packs must divide both eighteen and twenty-four. The greatest common factor is six, so make six packs. Each contains 18 ÷ 6 = 3 red pencils and 24 ÷ 6 = 4 blue pencils.
Notice that HCF here counts packs, not the number of items in each pack. The wording of the problem determines which common factor interpretation is useful.
Lowest common multiple
List multiples of six: 6, 12, 18, 24, 30, 36. List multiples of eight: 8, 16, 24, 32, 40. The first positive value common to both is 24.
LCM problems often involve repeated events occurring together again or finding the smallest shared grouping size.
Worked example B: Repeating schedules
One light flashes every six seconds and another every eight seconds. If they flash together now, the first time they flash together again is after the least common multiple of six and eight: 24 seconds.
The question asks when two repeating cycles coincide, so common multiples are the natural structure. HCF would answer a different kind of grouping question.
Prime-factor route for HCF and LCM
For 18 = 2 × 3 × 3 and 24 = 2 × 2 × 2 × 3, the common prime factors are one 2 and one 3, giving HCF 6.
For the LCM, include enough prime factors to build both numbers: three factors of 2 and two factors of 3, giving 8 × 9 = 72. Check that seventy-two is divisible by both eighteen and twenty-four.
Listing methods may be more transparent for small Primary numbers. Prime factorisation becomes efficient when the numbers are larger or several numbers are involved.
6. Practice: 24 questions
Keep the worked answers covered. For factor questions, use exact division or factor pairs. For common-factor and common-multiple questions, state why the context requires one rather than the other.
Questions 1–8: Factors and multiples
1. List all positive factors of 18.
2. Is 7 a factor of 42? Explain with a multiplication or division fact.
3. Is 5 a factor of 42?
4. Write the first six positive multiples of 8.
5. Which of 24, 25, 26 and 27 are multiples of 6?
6. Find the missing factor: 9 × __ = 63.
7. List all factor pairs of 36.
8. Explain the relationship between “6 is a factor of 24” and “24 is a multiple of 6.”
Questions 9–16: Divisibility and prime structure
9. Is 3,748 divisible by 2?
10. Is 7,316 divisible by 4?
11. Is 4,572 divisible by 9?
12. Which of 35, 40, 42 and 45 are divisible by 5?
13. Is 29 prime or composite?
14. Is 1 prime, composite or neither?
15. Write 36 as a product of prime factors.
16. Find the smallest prime factor of 91.
Questions 17–24: Common factors and multiples
17. Find the HCF of 18 and 24.
18. Find the LCM of 6 and 8.
19. Find the HCF of 30 and 45.
20. Find the LCM of 9 and 12.
21. Eighteen red pencils and twenty-four blue pencils are divided into the greatest possible number of identical packs with no leftovers. How many packs are made, and what is in each pack?
22. One bell rings every 12 minutes and another every 18 minutes. They ring together now. After how many minutes will they next ring together?
23. A number is divisible by both 4 and 6. What is the smallest positive number it could be?
24. Two numbers have HCF 6. Must both numbers be multiples of 6? Must their LCM equal 6? Explain.
7. Worked answers
Answers 1–8
1. 1, 2, 3, 6, 9, 18. The factor pairs are 1×18, 2×9 and 3×6. These use eighteen exactly.
2. Yes. 42 ÷ 7 = 6 with no remainder, or 7 × 6 = 42.
3. No. 42 ÷ 5 leaves remainder two, so five does not divide forty-two exactly.
4. 8, 16, 24, 32, 40, 48. These are 1×8 through 6×8.
5. 24 only. Twenty-four equals 6×4. The others are not exact multiples of six.
6. 7. Since 9×7 = 63, seven is the missing factor.
7. 1×36, 2×18, 3×12, 4×9, 6×6. Listing systematically prevents repeated or missing pairs.
8. They are two directions of the same multiplication relationship: 6×4=24. Six divides twenty-four exactly, and twenty-four is produced by a whole-number number of sixes.
Answers 9–16
9. Yes. The final digit is eight, an even digit, so the number is divisible by two.
10. Yes. The final two digits form sixteen, which is divisible by four.
11. Yes. The digit sum is 4+5+7+2=18, and eighteen is divisible by nine.
12. 35, 40 and 45. Numbers ending in zero or five are divisible by five. Forty-two is not.
13. Prime. Twenty-nine has no positive factors other than one and twenty-nine.
14. Neither. One has exactly one positive factor, so it fits neither definition.
15. 2×2×3×3. For example, 36=6×6 and each six is 2×3.
16. 7. Ninety-one is not divisible by two, three or five. It equals 7×13, so seven is its smallest prime factor.
Answers 17–24
17. 6. Common factors of eighteen and twenty-four are 1, 2, 3 and 6. The greatest is six.
18. 24. Multiples of six and eight first meet at twenty-four.
19. 15. Factors common to thirty and forty-five include 1, 3, 5 and 15. The greatest is fifteen.
20. 36. Multiples of nine are 9,18,27,36; multiples of twelve are 12,24,36. The first shared positive multiple is thirty-six.
21. Six packs; 3 red and 4 blue pencils per pack. The greatest number of packs must divide both eighteen and twenty-four. Their HCF is six.
22. 36 minutes. The first positive common multiple of twelve and eighteen is thirty-six.
23. 12. Twelve is divisible by both four and six and is the smallest positive common multiple.
24. Yes, both must be multiples of 6; no, the LCM need not equal 6. HCF six means six divides both. For example, twelve and eighteen have HCF six but LCM thirty-six.
8. Diagnose the relationship, not the vocabulary
If a learner confuses factors and multiples, return to one multiplication fact such as 4×6=24. Ask which numbers fit exactly inside twenty-four and which number is produced by repeated fours or sixes. Move between the two sentences until the direction is stable.
If divisibility rules are recalled but misapplied, verify with division on a few examples. The quick rule should predict exact division; it is not a separate game.
Use HCF and LCM only after the context is named
Making the greatest possible identical packs asks for a common divisor. Repeating cycles meeting again ask for a common multiple. Before computing, ask whether the answer is intended to divide quantities or be reached by repeated quantities.
This single distinction prevents many formula-selection errors.
Check prime claims with factors
A number is not prime because it “looks unusual.” Find or rule out factor pairs. If one non-trivial factor is found, the number is composite. If none exist within the necessary range, it is prime.
Connect to later mathematics
Factors support fraction simplification and algebraic factorisation. Multiples support common denominators and repeating cycles. Prime factorisation gives a compact description of divisibility structure. These later uses grow from the same equal-group idea.
Continue through the Primary Mathematics series
For the equal-group foundation, use Equal Groups, Division and Remainders. For fraction simplification and denominators, use Fractions, Decimals and the Same Whole. Continue to Patterns, Early Algebra and Equations.
Return to the BTT Primary Mathematics Learning Hub. The broader number concepts remain connected through the Number and Place Value and Arithmetic and Operations objects.
Original learning guide. Curriculum reference checked 6 September 2026. Examples and questions are original educational illustrations, not official examination items.