Factors, multiples and prime factorisation
12 chocolates can be shared equally among 1, 2, 3, 4, 6 or 12 friends. Those group sizes are the factors of 12.
A factor divides a number exactly; a multiple is in its times table. A prime has exactly two factors (1 and itself). Every number is a product of primes: its prime factorisation.
In the figure: Factor tree: split until only primes remain.
Factors, multiples and prime factorisation
In the app you can move this figure with your finger.
Open in the appRule 1: Find factors in pairs — Test 1, 2, 3 … and write the partner each time.
- Easy: Factors of 20: 1×20, 2×10, 4×5 → 6 factors.
- Medium: Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 → 9.
- Exam-type: 7 is a factor of 91 because 7 × 13 = 91.
Rule 2: Prime factorisation — Divide by 2, 3, 5, 7 … until you reach 1.
- Easy: 18 = 2 × 3 × 3 = 2 × 3².
- Medium: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
- Exam-type: 84 = 2² × 3 × 7.
Rule 3: Common factors and common multiples — List for both numbers and pick those that appear in both.
- Easy: Common factors of 16 and 24: 1, 2, 4, 8.
- Medium: Common multiples of 4 and 6: 12, 24, 36 …
- Exam-type: 1 is a common factor of every pair.
Every number has factors 1 and itself; 1 is not prime.
Calling 1 a prime number. A prime has exactly two factors.
Test divisibility by 2, 3, 5 first (end digit, digit sum). Memory clue: "Prime: exactly two factors."
Spot the difference
| Term | Is it ≤ or ≥ the number? |
|---|---|
| Factor | ≤ |
| Multiple | ≥ |
Practice questions
- 1How many factors does 20 have?
- 2Is 7 a factor of 91?
- 3Common factors of 16 and 24: how many?
- 4Prime factorisation of 84?
- 5How many factors does 36 have?
- 6What is the 4th multiple of 13?
- 7Smallest number with exactly 3 factors?
- 8Sum of the different prime factors of 210?
Show answers (try them yourself first)
- 16
- 2Yes
- 34
- 42² × 3 × 7
- 59
- 652
- 74
- 817
Highest Common Factor (HCF)
A 360 cm × 240 cm floor must be covered by square tiles with no cutting. The biggest tile that fits both lengths is their HCF.
The HCF of two or more numbers is the greatest number that divides all of them exactly. Find it by listing common factors or by taking the common primes with their smallest powers.
In the figure: Common factors of 12 and 18 are 1, 2, 3, 6; the greatest is 6.
Highest Common Factor (HCF)
In the app you can move this figure with your finger.
Open in the appRule 1: List the common factors — Pick the biggest factor present in both lists.
- Easy: 12: 1, 2, 3, 4, 6, 12; 18: 1, 2, 3, 6, 9, 18 → HCF 6.
- Medium: HCF(20, 30) = 10.
- Exam-type: HCF(15, 28) = 1.
Rule 2: Prime factor method — Multiply the primes common to all, each with its smallest power.
- Easy: 54 = 2 × 3³, 72 = 2³ × 3² → HCF = 2 × 3² = 18.
- Medium: 84 = 2²×3×7, 126 = 2×3²×7 → HCF = 2×3×7 = 42.
- Exam-type: HCF(24, 36, 60) = 12.
Rule 3: HCF as the biggest equal piece — Cutting two lengths into equal pieces of the longest size uses the HCF.
- Easy: Ribbons 45 m and 75 m: longest equal piece 15 m.
- Medium: Floor 360 × 240 cm: tile 120 cm.
- Exam-type: HCF of two different primes is 1.
HCF ≤ the smallest of the numbers.
Taking the largest power of each prime. For HCF take the smallest.
HCF = Highest (but Highest Common) — only primes in all numbers. Memory clue: "HCF: common primes, small powers."
Spot the difference
| Method | Powers |
|---|---|
| HCF | smallest |
| LCM | largest |
Practice questions
- 1HCF of 36, 48
- 2HCF of 54, 72
- 3HCF of 15, 28
- 4HCF of 84, 126
- 5HCF of 24, 36, 60
- 6HCF of 45, 75
- 7Floor 360 cm × 240 cm: side of the largest square tile (cm)?
- 8HCF of 144, 180, 216
Show answers (try them yourself first)
- 112
- 218
- 31
- 442
- 512
- 615
- 7120
- 836
Lowest Common Multiple (LCM)
Two buses leave a stand every 4 and every 6 minutes. After how many minutes do they leave together again? At the LCM.
The LCM of numbers is the smallest number that is a multiple of all of them. Find it by listing multiples or by taking every prime with its largest power.
In the figure: Multiples of 4 and 6 first meet at 12.
Lowest Common Multiple (LCM)
In the app you can move this figure with your finger.
Open in the appRule 1: List the multiples — The first number appearing in every list is the LCM.
- Easy: 4: 4, 8, 12 …; 6: 6, 12 … → LCM 12.
- Medium: LCM(5, 10) = 10.
- Exam-type: LCM(3, 7) = 21.
Rule 2: Prime factor method — Take every prime that appears, with its largest power.
- Easy: 18 = 2×3², 24 = 2³×3 → LCM = 2³×3² = 72.
- Medium: 12 = 2²×3, 15 = 3×5, 20 = 2²×5 → LCM = 2²×3×5 = 60.
- Exam-type: LCM(6, 8, 10) = 120.
Rule 3: LCM as the next meeting — Repeating events coincide again after the LCM of their gaps.
- Easy: Lights blink every 6 s and 8 s: together every 24 s.
- Medium: Smallest 4-digit multiple of 60 is 1020.
- Exam-type: LCM of coprime numbers is their product.
LCM ≥ the largest of the numbers.
Giving the product 4 × 6 = 24 as LCM. Divide out the common part: 12.
Check: the LCM must divide evenly by each number. Memory clue: "LCM: all primes, big powers."
Spot the difference
| Question type | Use |
|---|---|
| Biggest equal pieces | HCF |
| Meet again | LCM |
Practice questions
- 1LCM of 4, 6
- 2LCM of 5, 10
- 3LCM of 8, 12
- 4LCM of 18, 24
- 5LCM of 12, 15, 20
- 6LCM of 6, 8, 10
- 7Bells ring every 12 and 18 min. After how many minutes do they ring together?
- 8LCM of 28, 42, 63
Show answers (try them yourself first)
- 112
- 210
- 324
- 472
- 560
- 6120
- 736
- 8252
Division method and HCF × LCM
To find the biggest tile for large lengths like 1071 and 462, listing factors is too slow. A quick division ladder does it in three steps.
In the division method divide the larger number by the smaller, then divide the divisor by the remainder, and so on. The last non-zero remainder-divisor is the HCF. For two numbers, HCF × LCM = product of the numbers. Numbers with HCF 1 are coprime.
In the figure: Divide again and again until the remainder is 0.
Division method and HCF × LCM
In the app you can move this figure with your finger.
Open in the appRule 1: Division steps — Larger = quotient × smaller + remainder; repeat with (smaller, remainder).
- Easy: HCF(48, 18): 48 = 2×18 + 12; 18 = 1×12 + 6; 12 = 2×6 → 6.
- Medium: HCF(105, 45): 105 = 2×45 + 15; 45 = 3×15 → 15.
- Exam-type: HCF(1071, 462) = 21.
Rule 2: HCF × LCM = product — Use it to find a missing number.
- Easy: 12 and 18: HCF 6, LCM 36, 6 × 36 = 216 = 12 × 18.
- Medium: HCF 4, LCM 60, one number 12 → other 4×60÷12 = 20.
- Exam-type: HCF 9, LCM 90, one number 18 → other 45.
Rule 3: Coprime numbers — Two numbers are coprime when their only common factor is 1.
- Easy: 8 and 15 are coprime.
- Medium: Coprime numbers need not be prime: 14 and 25.
- Exam-type: Consecutive numbers are always coprime.
HCF × LCM = product works for two numbers only.
Using it for three numbers. It does not hold there.
Always divide the divisor by the remainder next. Memory clue: "Remainder becomes the divisor."
Spot the difference
| Pair | HCF | LCM |
|---|---|---|
| 8, 15 | 1 | 120 |
| 12, 18 | 6 | 36 |
Practice questions
- 1HCF of 48 and 18 by division?
- 2HCF of 105 and 45?
- 3Are 8 and 15 coprime?
- 4HCF 6 and LCM 36: product of the two numbers?
- 5HCF 4, LCM 60, one number 12. The other?
- 6HCF of 119 and 34?
- 7HCF of 1071 and 462?
- 8HCF 9, LCM 90, one number 18. The other?
Show answers (try them yourself first)
- 16
- 215
- 3Yes
- 4216
- 520
- 617
- 721
- 845
HCF and LCM word problems
Cutting ropes into the longest equal pieces, making teams, tiling a floor, or timing buses: each hides an HCF or an LCM.
Ask: do I need the biggest equal share (HCF) or the first time things match again (LCM)? For remainders, subtract the remainder first, then use HCF, or take the LCM and add the remainder.
In the figure: Floor 240 × 120: largest square tile 60 (the HCF).
HCF and LCM word problems
In the app you can move this figure with your finger.
Open in the appRule 1: HCF problems — Words: greatest, largest, equal pieces, groups, tiles.
- Easy: Ropes 48 m and 72 m: longest equal piece 24 m.
- Medium: 24 boys and 36 girls: most equal teams 12.
- Exam-type: Room 420 × 300 cm: tile 60 cm, tiles (420÷60)×(300÷60) = 35.
Rule 2: LCM problems — Words: together again, smallest, least number, after how long.
- Easy: Buses every 15 and 20 min: together again after 60 min.
- Medium: Bells every 10, 12, 15 min: together after 60 min.
- Exam-type: Smallest 3-digit multiple of 12 and 15 is 120.
Rule 3: Remainder problems — Same remainder for all divisors: LCM + remainder. Divides leaving remainder r: HCF of (number – r).
- Easy: Leaves 1 when divided by 4 and 6: 12 + 1 = 13.
- Medium: Leaves 5 on 6, 8, 12: 24 + 5 = 29.
- Exam-type: Divides 85 and 134 leaving 1: HCF(84, 133) = 7.
Greatest → HCF; least or together again → LCM.
Adding the remainder in HCF problems. There you subtract it first.
Underline the key words before choosing. Memory clue: "Big share: HCF. Meet again: LCM."
Spot the difference
| Key words | Use |
|---|---|
| largest, equal pieces | HCF |
| least, together again | LCM |
Practice questions
- 1Ropes 48 m and 72 m: longest equal piece (m)?
- 2Lights blink every 6 s and 8 s. Together after (s)?
- 324 boys and 36 girls: maximum equal teams?
- 4Smallest number leaving remainder 1 on division by 4 and 6 (more than 1)?
- 5Smallest 3-digit multiple of 12 and 15?
- 6Room 420 × 300 cm with 60 cm tiles: number of tiles?
- 7Smallest number leaving remainder 5 on division by 6, 8 and 12?
- 8Largest number dividing 85 and 134 leaving remainder 1 each?
Show answers (try them yourself first)
- 124
- 224
- 312
- 413
- 5120
- 635
- 729
- 87