Multiples, factors and what they share
Two bells ring every 6 and every 8 minutes. When will they ring together? A shopkeeper packs 42 pens and 70 pencils into equal packets. How many packets? Both questions are about common multiples and common factors.
A multiple of a number is what we get by multiplying it by 1, 2, 3, …. A factor divides the number exactly. 1 is a factor of every number, and every number is a factor of itself. A common multiple is a multiple of both numbers; a common factor divides both. A perfect number equals the sum of its factors other than itself, for example 6 and 28.
In the figure: The factors of 12 are 1, 2, 3, 4, 6 and 12. They come in pairs: 1 × 12, 2 × 6 and 3 × 4.
Rule 1: Find multiples and factors — Multiples: keep adding the number (6, 12, 18, …). Factors: try dividing by 1, 2, 3, … and keep the numbers that divide exactly. Factors come in pairs, so you can stop when the pairs repeat.
- Easy: Factors of 18: 1, 2, 3, 6, 9, 18 (from 1 × 18, 2 × 9, 3 × 6).
- Medium: Multiples of 7: 7, 14, 21, 28, 35. So 35 is the 5th multiple, since 7 × 5 = 35.
- Exam-type: Is 24 a multiple of 8? Yes, 8 × 3 = 24. Factors of 20: 1, 2, 4, 5, 10, 20, which is 6 factors.
Rule 2: Common multiples and common factors — Write both lists. The first number that appears in both lists of multiples is the smallest common multiple. The largest number in both lists of factors is the greatest common factor.
Multiples, factors and what they share
In the app you can move this figure with your finger.
Open in the app- Easy: Multiples of 4: 4, 8, 12. Multiples of 6: 6, 12. The first common multiple is 12.
- Medium: Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. Common: 1, 2, 3, 6; the greatest is 6.
- Exam-type: Bells every 6 and 8 minutes ring together after 24 minutes. For 42 pens and 70 pencils the greatest number of packets is 14, since 42 = 14 × 3 and 70 = 14 × 5.
Every multiple is at least as big as the number; every factor is at most as big. 28 = 1 + 2 + 4 + 7 + 14, so 28 is perfect.
Mixing up the two words: factors divide the number, multiples are divisible by it.
Multiples go up by adding; factors come in pairs by dividing. Memory clue: “Multiples Multiply upward, Factors Fit inside”.
Spot the difference
| Word | Meaning | Example |
|---|---|---|
| Factor | divides exactly | 1, 2, 3, 4, 6, 12 for 12 |
| Multiple | number × 1, 2, 3, … | 12, 24, 36, … |
| Common multiple | multiple of both | 12 for 4 and 6 |
| Common factor | divides both | 6 for 12 and 18 |
Practice questions
- 1Name a factor of every number.
- 2What is the third multiple of 6?
- 3Write all the factors of 10.
- 4What is the smallest number above 0 that is a multiple of both 4 and 6?
- 5What is the greatest common factor of 12 and 18?
- 6How many factors does 20 have?
- 7What is the smallest number that is a multiple of both 12 and 15?
- 8Three lights blink every 4, 6 and 10 seconds and blink together now. After how many seconds will they first blink together again?
Show answers (try them yourself first)
- 11) 1
- 22) 18
- 33) 1, 2, 5, 10
- 44) 12
- 55) 6
- 66) 6 (1, 2, 4, 5, 10, 20)
- 77) 60
- 88) 60 seconds (the smallest common multiple of 4, 6 and 10)
Prime numbers, composite numbers and the Sieve
Try to share 7 chocolates equally among 2, 3, 4, 5 or 6 friends: it never works without leftovers. But 12 chocolates can be shared in many ways. Numbers like 7 that cannot be shared in groups are called prime.
A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither. 2 is the only even prime. Twin primes are primes that differ by 2, like 11 and 13. The Sieve of Eratosthenes finds primes by crossing out multiples.
In the figure: The 15 primes up to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.
Rule 1: Prime or composite? — Try dividing by 2, 3, 5, 7, …. If any number other than 1 and the number itself divides it, it is composite. For a small number it is enough to try primes p with p × p not bigger than the number.
- Easy: 13 is divisible only by 1 and 13, so it is prime. 21 = 3 × 7, so it is composite.
- Medium: 91 = 7 × 13, so it is composite even though it looks prime. 53 has no factor except 1 and 53, so it is prime.
- Exam-type: 9 has exactly three factors, 1, 3 and 9. 2 is the only even prime, because every other even number also has the factor 2.
Rule 2: Use the Sieve of Eratosthenes — Cross out 1. Take 2 and cross out all its multiples. Take the next number not crossed out, 3, and cross out its multiples. Repeat. Stop when p × p is bigger than the last number; everything left is prime.
Prime numbers, composite numbers and the Sieve
In the app you can move this figure with your finger.
Open in the app- Easy: Up to 10 the primes are 2, 3, 5 and 7: four primes.
- Medium: Between 10 and 20 the primes are 11, 13, 17 and 19 (4 primes). Between 20 and 30 they are 23 and 29 (2 primes).
- Exam-type: For 1 to 50 we only cross out multiples of 2, 3, 5 and 7, since 7 × 7 = 49 is below 50 but 11 × 11 = 121 is above. Twin primes: 11 and 13, 17 and 19.
1 is neither prime nor composite. 2 is the smallest prime and the only even one.
Calling 1 a prime, or thinking every odd number is prime. 9, 15, 21 and 91 are odd but composite.
No even number above 2 and no number above 5 ending in 5 is prime. Memory clue: “Prime = Two Factors Only”.
Spot the difference
| Number | Factors | Kind |
|---|---|---|
| 1 | 1 | neither |
| 2 | 1, 2 | prime (the only even prime) |
| 9 | 1, 3, 9 | composite |
| 13 | 1, 13 | prime |
Practice questions
- 1What is the smallest prime number?
- 2Which is prime: 9, 13 or 15?
- 3Is 1 a prime number?
- 4How many primes are there between 10 and 20?
- 5Which of 51, 53, 57 and 87 is prime?
- 6Which pair is twin primes: 13 and 17, 17 and 19, or 7 and 11?
- 7To find all primes up to 50 with the Sieve, what is the largest prime whose multiples must be crossed out?
- 8Which number between 90 and 100 is prime?
Show answers (try them yourself first)
- 11) 2
- 22) 13 (9 = 3 × 3 and 15 = 3 × 5)
- 33) No, 1 is neither prime nor composite
- 44) 4 (11, 13, 17, 19)
- 55) 53 (the others are multiples of 3)
- 66) 17 and 19
- 77) 7 (7 × 7 = 49 < 50 but 11 × 11 = 121 > 50)
- 88) 97 (91 = 7 × 13, 93 = 3 × 31, 99 = 9 × 11)
Co-prime numbers
Two toothed wheels with 9 and 14 teeth turn together. Because 9 and 14 share no factor except 1, the same two teeth meet again only after 9 × 14 = 126 teeth have passed.
Two numbers are co-prime if their only common factor is 1. The numbers themselves need not be prime: 4 and 9 are co-prime. Any two different primes are co-prime, 1 is co-prime with every number, and neighbouring numbers are co-prime.
In the figure: 14 = 2 × 7 and 15 = 3 × 5 share no prime, so they are co-prime. 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3 share 2 and 3, so they are not.
Rule 1: Test with prime factors — Write the prime factors of both numbers. If they share no prime, the numbers are co-prime, and their greatest common factor is 1.
Co-prime numbers
In the app you can move this figure with your finger.
Open in the app- Easy: 4 and 9: factors 1, 2, 4 and 1, 3, 9. Only 1 is common, so they are co-prime.
- Medium: 33 = 3 × 11 and 35 = 5 × 7 share no prime: co-prime. 33 and 36 share 3: not co-prime.
- Exam-type: Numbers from 1 to 20 co-prime with 20: drop multiples of 2 and 5, leaving 1, 3, 7, 9, 11, 13, 17, 19. That is 8 numbers.
Rule 2: Quick facts about co-prime numbers — Different primes are always co-prime. Neighbouring numbers are co-prime. Two even numbers are never co-prime, because both have the factor 2. For co-prime numbers the smallest common multiple is their product.
- Easy: 7 and 11 are co-prime (different primes). 14 and 15 are co-prime (neighbours).
- Medium: 8 and 15 are co-prime, so their smallest common multiple is 8 × 15 = 120.
- Exam-type: 4 and 9 are co-prime, yet neither is prime. So co-prime numbers need not be prime.
Co-prime describes a pair of numbers, not a single number.
Thinking co-prime numbers must themselves be prime, or that 1 cannot be co-prime with anything.
No shared prime means co-prime. Memory clue: “Share nothing but 1”.
Spot the difference
| Pair | Common factors | Co-prime? |
|---|---|---|
| 4 and 9 | 1 | yes |
| 6 and 9 | 1, 3 | no |
| 14 and 15 | 1 | yes |
| 12 and 18 | 1, 2, 3, 6 | no |
Practice questions
- 1What is the only common factor of 5 and 8?
- 2Which pair is co-prime: 4 and 6, 4 and 9, or 8 and 12?
- 3Are any two different primes always co-prime?
- 4Which of 4, 5, 8 and 9 is co-prime with 10?
- 5What is the smallest common multiple of the co-prime numbers 8 and 15?
- 6How many numbers from 1 to 10 are co-prime with 10?
- 7How many numbers from 1 to 20 are co-prime with 20?
- 8Which of 24, 25, 26 and 27 is co-prime with 12?
Show answers (try them yourself first)
- 11) 1
- 22) 4 and 9
- 33) Yes, they share no factor except 1
- 44) 9
- 55) 120 (8 × 15)
- 66) 4 (1, 3, 7, 9)
- 77) 8 (1, 3, 7, 9, 11, 13, 17, 19)
- 88) 25
Prime factorisation
A builder says a wall uses 60 bricks. Breaking 60 into smaller and smaller equal parts, 60 = 6 × 10 = (2 × 3) × (2 × 5), ends with bricks that cannot be split any further. Those unbreakable parts are primes.
Writing a number as a product of primes is its prime factorisation. Every number above 1 has exactly one prime factorisation, apart from the order of the factors. A factor tree shows the splitting step by step.
In the figure: 60 splits into 6 and 10, then into 2, 3, 2 and 5. So 60 = 2 × 2 × 3 × 5.
Rule 1: Split until only primes are left — Divide by the smallest prime that fits (2, then 3, then 5, …) and keep going with the quotient until it becomes 1. Or draw a factor tree and stop each branch when it reaches a prime. Multiply the primes back to check.
Prime factorisation
In the app you can move this figure with your finger.
Open in the app- Easy: 84: 84 ÷ 2 = 42, 42 ÷ 2 = 21, 21 ÷ 3 = 7. So 84 = 2 × 2 × 3 × 7. Check: 4 × 3 × 7 = 84.
- Medium: 100 = 10 × 10 = (2 × 5) × (2 × 5) = 2 × 2 × 5 × 5. A prime such as 13 is already its own factorisation.
- Exam-type: 126 = 2 × 63 = 2 × 3 × 21 = 2 × 3 × 3 × 7. A different tree, such as 126 = 9 × 14, gives the same primes.
Rule 2: Use prime factors for common factor and multiple — Write both factorisations. The common primes, taken together, multiply to the greatest common factor. All the primes of both, with the shared ones counted once, multiply to the smallest common multiple.
- Easy: 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. Common 2 × 3 = 6, so the greatest common factor is 6. Smallest common multiple: 2 × 2 × 3 × 3 = 36.
- Medium: 20 = 2 × 2 × 5 and 30 = 2 × 3 × 5. Greatest common factor 2 × 5 = 10. Smallest common multiple 2 × 2 × 3 × 5 = 60.
- Exam-type: 8 = 2 × 2 × 2 and 9 = 3 × 3 share no prime, so they are co-prime. The smallest common multiple is 8 × 9 = 72.
Always check by multiplying the primes back. Only primes may appear in the final line.
Stopping too early, e.g. writing 36 = 4 × 9. Here 4 and 9 are not prime; go on to 2 × 2 × 3 × 3.
Start with 2 as many times as it fits, then 3, then 5. Memory clue: “Break, break, until only primes remain”.
Spot the difference
| Number | Prime factorisation | Check |
|---|---|---|
| 12 | 2 × 2 × 3 | 4 × 3 = 12 |
| 60 | 2 × 2 × 3 × 5 | 12 × 5 = 60 |
| 100 | 2 × 2 × 5 × 5 | 4 × 25 = 100 |
| 13 | 13 (prime) | 13 |
Practice questions
- 1Write 12 as a product of primes.
- 2What are the prime factors of 15?
- 3Is 2 × 3 × 4 a prime factorisation? Why or why not?
- 4Write the prime factorisation of 60.
- 5Which number has the prime factorisation 2 × 3 × 7?
- 6Use prime factors to find the greatest common factor of 20 and 30.
- 7Write the prime factorisation of 144.
- 8Use prime factorisation to find the greatest common factor and the smallest common multiple of 24 and 36.
Show answers (try them yourself first)
- 11) 2 × 2 × 3
- 22) 3 and 5
- 33) No, 4 is not prime (2 × 3 × 2 × 2 is the correct form)
- 44) 2 × 2 × 3 × 5
- 55) 42
- 66) 10 (20 = 2 × 2 × 5, 30 = 2 × 3 × 5)
- 77) 2 × 2 × 2 × 2 × 3 × 3 (16 × 9 = 144)
- 88) Greatest common factor 12, smallest common multiple 72 (24 = 2 × 2 × 2 × 3, 36 = 2 × 2 × 3 × 3)
Divisibility tests for 2, 4, 5, 8 and 10
A trader packs 1 316 eggs into trays of 4. Does he need a calculator to know if any will be left over? No: a quick look at the last digits of the number is enough.
A number is divisible by another if the division leaves no remainder. Short divisibility tests let us decide this just by looking at the last digit, last two digits or last three digits of the number.
In the figure: 2, 5 and 10 look at the last digit; 4 looks at the last two digits; 8 looks at the last three digits.
Rule 1: Tests for 10, 5 and 2: the last digit — Divisible by 10 if the last digit is 0. Divisible by 5 if the last digit is 0 or 5. Divisible by 2 if the last digit is 0, 2, 4, 6 or 8.
Divisibility tests for 2, 4, 5, 8 and 10
In the app you can move this figure with your finger.
Open in the app- Easy: 340 ends in 0, so it is divisible by 10, by 5 and by 2.
- Medium: 275 ends in 5: divisible by 5, but not by 2 or 10.
- Exam-type: 128 ends in 8: divisible by 2, but not by 5 or 10.
Rule 2: Tests for 4 and 8: last two and last three digits — Divisible by 4 if the number formed by the last two digits is divisible by 4 (00 also works). Divisible by 8 if the number formed by the last three digits is divisible by 8 (000 also works).
- Easy: 1 316: last two digits 16, and 16 ÷ 4 = 4. So 1 316 is divisible by 4, and the trays fill exactly.
- Medium: 2 750: last two digits 50, and 50 ÷ 4 leaves 2. So 2 750 is not divisible by 4.
- Exam-type: 5 240: last three digits 240, and 240 ÷ 8 = 30. So 5 240 is divisible by 8. For 3 124, 124 ÷ 8 leaves 4, so it is not divisible by 8, though 24 ÷ 4 = 6 makes it divisible by 4.
2, 4 and 8 are 2, 2 × 2 and 2 × 2 × 2: look at 1, 2 and 3 last digits. A number divisible by 8 is also divisible by 4 and 2.
Looking at the first digits instead of the last, or testing 4 using only the last digit. 1 314 ends in 4 but 14 is not divisible by 4.
Last digit for 2, 5, 10; last two for 4; last three for 8. Memory clue: “2, 4, 8 are powers of 2: look at 1, 2, 3 digits from the tail”.
Spot the difference
| Divisor | What to look at | Example |
|---|---|---|
| 2 | last digit 0, 2, 4, 6, 8 | 358 |
| 5 and 10 | last digit 0 or 5 and last digit 0 | 465 and 730 |
| 4 | last two digits divisible by 4 | 1 316 |
| 8 | last three digits divisible by 8 | 5 240 |
Practice questions
- 1Is 470 divisible by 10?
- 2Is 625 divisible by 2?
- 3Which last digits can a multiple of 5 have?
- 4Is 1 348 divisible by 4?
- 5Is 3 250 divisible by 8?
- 6Which of 135, 240 and 316 is divisible by both 2 and 5?
- 7Which digit in the box makes 38□ divisible by 8?
- 8What is the smallest number above 100 that is divisible by both 4 and 5?
Show answers (try them yourself first)
- 11) Yes, it ends in 0
- 22) No, it ends in 5 (odd)
- 33) 0 or 5
- 44) Yes (48 ÷ 4 = 12)
- 55) No (250 ÷ 8 leaves 2)
- 66) 240 (it ends in 0)
- 77) 4 (384 ÷ 8 = 48)
- 88) 120 (it must be a multiple of 20)