Cambridge Primary Checkpoint · Mathematics 0096 · Stage 6 · Number
Factors, Multiples and Tests of Divisibility
Find factors, multiples, common factors and common multiples, and use quick tests to see whether a number is divisible by 3, 6 or 9.
British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course
- Understand common multiples and common factors of two numbers, and use them to solve problems (6Ni.06).
- Use what you know about factors and multiples to understand and use the tests of divisibility by 3, 6 and 9 (6Ni.07).
Factors and multiples are the "family members" of a number. They help you simplify fractions, find common denominators, check divisions and solve real problems such as "When will the two buses leave together again?". On this page you will also learn quick tests that tell you if a big number can be divided by 3, 6 or 9 — without dividing!
Key words
| Word / phrase | Meaning / use |
|---|---|
| factor | عامل — a whole number that divides another number exactly. 4 is a factor of 24. |
| factor pair | زوج عوامل — two factors that multiply to give the number: 4 × 6 = 24 |
| multiple | مضاعف — a number in the times table of another number. 24 is a multiple of 4. |
| common factor | عامل مشترك — a number that is a factor of both numbers |
| highest common factor (HCF) | العامل المشترك الأكبر — the biggest common factor |
| common multiple | مضاعف مشترك — a number that is a multiple of both numbers |
| lowest common multiple (LCM) | المضاعف المشترك الأصغر — the smallest common multiple |
| divisible by | يقبل القسمة على — can be divided exactly (no remainder) |
| digit sum | مجموع الأرقام — add all the digits: the digit sum of 4 734 is 4 + 7 + 3 + 4 = 18 |
Factor or multiple? Factors are small (they fit into the number): factors of 12 are 1, 2, 3, 4, 6, 12. Multiples are big (the number fits into them) and go on for ever: multiples of 12 are 12, 24, 36, 48, … The two ideas are linked: "4 is a factor of 24" means the same as "24 is a multiple of 4".
1. Finding all the factors of a number
Work in pairs so you never miss one.
- Start with 1 × the number: 1 × 36.
- Try 2, 3, 4, 5, … in order. If it divides exactly, write the pair: 2 × 18, 3 × 12, 4 × 9. (5 does not divide 36.)
- Stop when the pairs meet or repeat: 6 × 6. The next one, 9 × 4, is already written.
- List the factors in order: 1, 2, 3, 4, 6, 9, 12, 18, 36.
2. Common factors
List the factors of both numbers, then find the numbers that appear in both lists. A Venn diagram shows this clearly: common factors go in the overlap.
1 is a common factor of every pair of numbers. The highest common factor is useful for simplifying fractions: 24/36 — divide top and bottom by 12 → 2/3.
3. Common multiples
List the first few multiples of each number until you find numbers in both lists.
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, …
- Common multiples of 4 and 6: 12, 24, 36, … The lowest common multiple is 12. Every other common multiple is a multiple of 12.
Quick method for the lowest common multiple: go through the multiples of the bigger number and stop at the first one the smaller number divides. LCM of 6 and 8: 8 (no), 16 (no), 24 (yes, 24 ÷ 6 = 4). Careful: the LCM is not always the two numbers multiplied: 6 × 8 = 48 is a common multiple, but not the lowest.
| Problem says… | Usually means | Example |
|---|---|---|
| "together again", "at the same time", "first time both…" | lowest common multiple | Buses every 12 and 18 minutes leave together; again after LCM = 36 minutes |
| "largest possible equal pieces / groups", "no waste", "share both equally" | highest common factor | Ribbons 24 cm and 40 cm cut into equal pieces as long as possible: HCF = 8 cm |
4. Tests of divisibility
You already know some tests from earlier stages. Stage 6 adds the tests for 3, 6 and 9.
| Divisible by | Test | Example |
|---|---|---|
| 2 | the last digit is even (0, 2, 4, 6, 8) | 4 734 ✓ |
| 5 | the last digit is 0 or 5 | 7 425 ✓ |
| 10 | the last digit is 0 | 6 250 ✓ |
| 4 | the last two digits make a multiple of 4 | 3 516 ✓ (16 = 4 × 4) |
| 3 | the digit sum is a multiple of 3 | 2 316: 2 + 3 + 1 + 6 = 12 ✓ |
| 9 | the digit sum is a multiple of 9 | 7 425: 7 + 4 + 2 + 5 = 18 ✓ |
| 6 | the number is even AND passes the test for 3 | 4 734: even ✓, 4 + 7 + 3 + 4 = 18 ✓ |
If the digit sum is still big, add its digits again: 5 976 → 5 + 9 + 7 + 6 = 27 → 2 + 7 = 9, so 5 976 is divisible by 9 (and by 3).
Why does the digit sum work? 10 = 9 + 1, 100 = 99 + 1, 1000 = 999 + 1. So 432 = 4 × 100 + 3 × 10 + 2 = (4 × 99 + 3 × 9) + (4 + 3 + 2). The part in the first bracket is always a multiple of 9 (and so of 3). So 432 is divisible by 9 exactly when the digit sum 4 + 3 + 2 = 9 is. This is a good example of convincing — explaining why a rule is always true.
Why does the test for 6 need two parts? 6 = 2 × 3, so a multiple of 6 must be a multiple of 2 and of 3. 15 passes the test for 3 but is odd, so it is not divisible by 6. 14 is even but its digit sum is 5, so it is not divisible by 6.
- Every multiple of 9 is a multiple of 3 (9 = 3 × 3). But not every multiple of 3 is a multiple of 9: 12, 15, 21 are counter-examples.
- Every multiple of 6 is even and a multiple of 3.
- A number with digit sum 9, 18, 27 … is divisible by 3 and 9.
- القسمة على 3: اجمع أرقام العدد، لو المجموع من جدول الـ 3 يبقى العدد يقبل القسمة على 3.
- القسمة على 9: نفس الفكرة، بس المجموع لازم يكون من جدول الـ 9 (9، 18، 27…).
- القسمة على 6: لازم الشرطين مع بعض — العدد زوجي (آخر رقم زوجي) ومجموع أرقامه من جدول الـ 3.
- العوامل صغيرة وبتدخل في العدد، والمضاعفات كبيرة ومالهاش نهاية.
Worked examples
Worked example 1 (1 mark) — Write all the factors of 48.
Pairs: 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8 (5 and 7 do not divide 48; the next, 8 × 6, repeats). Answer: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Common mistake: forgetting 1 and 48 themselves.
Worked example 2 (1 mark) — Find the highest common factor of 18 and 30.
Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. Common: 1, 2, 3, 6. Answer: 6.
Worked example 3 (1 mark) — Write the two common multiples of 6 and 8 that are less than 50.
Multiples of 8: 8, 16, 24, 32, 40, 48. Which are divisible by 6? 24 and 48. Answer: 24 and 48.
Worked example 4 (1 mark) — Tick the numbers that are divisible by 6: 3 582 7 425 2 316 4 105
| Number | Even? | Digit sum | Divisible by 3? | Divisible by 6? |
|---|---|---|---|---|
| 3 582 | yes | 18 | yes | yes ✓ |
| 7 425 | no | 18 | yes | no (odd) |
| 2 316 | yes | 12 | yes | yes ✓ |
| 4 105 | no | 10 | no | no |
Common mistake: ticking 7 425 because its digit sum is 18. It is divisible by 3 and by 9, but it is odd, so it is not divisible by 6.
Worked example 5 (2 marks) — 73□ is a 3-digit number that is divisible by 6. Find all the possible missing digits.
It must be even: □ is 0, 2, 4, 6 or 8. Digit sum 7 + 3 + □ = 10 + □ must be a multiple of 3: □ = 2 (sum 12) or □ = 8 (sum 18). Both are even. Answer: 2 or 8 (732 and 738). Check: 732 ÷ 6 = 122 ✓, 738 ÷ 6 = 123 ✓. Common mistake: also giving 5 (sum 15) — but 735 is odd.
Worked example 6 (1 mark) — Two lights flash at the same moment. One flashes every 8 seconds, the other every 12 seconds. After how many seconds will they next flash together?
Multiples of 12: 12, 24. 24 ÷ 8 = 3 ✓. Answer: 24 seconds (the lowest common multiple). Common mistake: 8 × 12 = 96 — that is a common multiple, but not the first one.
- Mixing up factors and multiples. Factors of 12 are ≤ 12; multiples of 12 are ≥ 12.
- Missing factor pairs — always work systematically from 1 upwards.
- Forgetting 1 and the number itself as factors.
- Using the digit-sum test for 6 without checking the number is even.
- Thinking "divisible by 3" means "divisible by 9" — 12 is divisible by 3 but not 9.
- Giving a common multiple that is not the lowest (multiplying the two numbers).
- Using the last digit to test for 3 or 9 — those tests need the digit sum.
- "Write all the factors of …" / "Write a common factor of …" — list in order, in pairs.
- "Write a number that is a multiple of 6 and a multiple of 9" — any common multiple (18, 36, 54 …) gets the mark.
- Venn and Carroll diagrams: sort numbers by "factor of 24", "multiple of 3", "divisible by 6" … Check every number against both labels.
- "Tick the numbers that are divisible by 9" — use the digit sum, write it next to each number.
- Missing-digit questions: "□ is a digit. 5□4 is divisible by 9. Find the possible values." — there may be more than one answer.
- "Is Sara correct? Explain." — give a counter-example (one number that breaks the rule) or a test result.
- Word problems: "together again" = LCM; "largest equal groups / pieces" = HCF.
Practice
- Find factors in pairs from 1 upwards; multiples are the times table and never end.
- Common factor = in both factor lists (HCF = biggest); common multiple = in both multiple lists (LCM = smallest).
- Divisible by 3 → digit sum is a multiple of 3. Divisible by 9 → digit sum is a multiple of 9.
- Divisible by 6 → even AND digit sum a multiple of 3.
- "Together again" → LCM. "Largest equal pieces/groups" → HCF.
All pages in this course
Start here
Number: integers and calculation
- Counting On and Back and Negative Numbers
- Sequences, Letters and Position-to-Term Rules
- Order of Operations, Brackets and Laws of Arithmetic
- Multiplying Whole Numbers (up to 10 000 × 2-digit)
- Dividing Whole Numbers (up to 1000 ÷ 2-digit)
- Factors, Multiples and Tests of Divisibility
- Square Numbers and Cube Numbers
Number: place value, decimals, fractions, percentages and ratio
- Place Value and Decimals to Thousandths
- Rounding Decimals
- Fractions: Division, Operators and Simplest Form
- Adding and Subtracting Fractions
- Multiplying and Dividing Fractions by Whole Numbers
- Fractions, Decimals and Percentages: Equivalence and Ordering
- Adding, Subtracting, Multiplying and Dividing Decimals
- Ratio and Direct Proportion
Geometry and measure
- Time: Decimals and Mixed Units
- Quadrilaterals: Properties and Classification
- Circles: Parts and Constructing Circles
- Angles: Measuring, Drawing and Angles in a Triangle
- Area of Rectangles and Right-angled Triangles
- 3D Shapes, Compound Shapes and Nets
- Capacity, Volume and Surface Area
- Coordinates in Four Quadrants
- Rotational Symmetry, Translation, Reflection and Rotation
Statistics and probability
Practice tests