Year 5: Prime and composite numbers
Part of the Year 5 prime numbers, factors, squares and cubes chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Prime and composite numbers
Look at the factors of 7: only 1 and 7. Look at the factors of 8: 1, 2, 4 and 8. A number like 7 is special. It is called a prime number. A prime number has exactly two factors: 1 and itself. A number like 8 has more than two factors, so it is a composite number. With square tiles you can see the difference. A prime can only make one rectangle: a single long row. A composite number can make more than one.
Two special numbers. The number 1 has only one factor, so it is neither prime nor composite. The number 2 is prime (factors 1 and 2), and it is the only even prime, because every other even number has 2 as a factor as well as 1 and itself.
Learn the primes up to 19: 2, 3, 5, 7, 11, 13, 17 and 19.
The Sieve of Eratosthenes. This is an old way to find primes, using a hundred square. Cross out 1. Keep 2, then cross out every multiple of 2. Keep 3, then cross out every multiple of 3. Keep 5 and cross out its multiples. Keep 7 and cross out its multiples. All the numbers that are left are the primes.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
| 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
| 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
| 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
| 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
Is a number up to 100 prime? You do not need a sieve every time. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them divides the number, it is not prime. If none of them does, the number is prime. Why only these four? Because 11 × 11 = 121 is bigger than 100, so any number up to 100 that is not prime must have a factor of 2, 3, 5 or 7.
Key ideas
- A prime number has exactly two factors: 1 and itself. A composite number has more than two factors.
- 1 is neither prime nor composite. 2 is the only even prime.
- Primes up to 19: 2, 3, 5, 7, 11, 13, 17, 19. There are 25 primes up to 100.
- To test a number up to 100: use the tests for 2, 3, 5 and 7. If none works, it is prime.
- The Sieve of Eratosthenes crosses out the multiples of 2, 3, 5 and 7. The numbers left are prime.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find all the factors, count them, and decide.
- List the factors of 13. Use factor pairs.1 × 13
Factors: 1, 13 - Count the factors. 2 factors
- Decide. Prime: exactly two factors. Composite: more than two. 1: only one.13 is prime. Exactly two factors, so prime.
Check. Every number except 1 is either prime or composite. 13 has 2 factors. ✓
Watch out. 1 is not prime. 2 is prime (it has exactly two factors).
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find all the factors, count them, and decide.
- List the factors of 21. Use factor pairs.1 × 21, 3 × 7
Factors: 1, 3, 7, 21 - Count the factors. 4 factors
- Decide. Prime: exactly two factors. Composite: more than two. 1: only one.21 is composite. More than two factors, so composite.
Check. Every number except 1 is either prime or composite. 21 has 4 factors. ✓
Watch out. 1 is not prime. 2 is prime (it has exactly two factors).
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. A prime number has exactly two factors. Find the factors of 1.
- List the factors of 1. Only 1 divides into 1.Factors of 1: 1
- Count the factors. A prime needs exactly two different factors.1 has only one factor.
- Decide. A composite number has more than two factors, so 1 is not composite either.1 is neither prime nor composite.
Check. The first prime is 2, with factors 1 and 2. ✓
Watch out. 1 is not a prime number, even though it looks like one.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.
- Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.We look for a factor that is not 1 or 51
- Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.Is it divisible by 2? The last digit of 51 is 1. Is it 0, 2, 4, 6 or 8? no.
Is it divisible by 3? 5 + 1 = 6. 6 is in the 3 times table (3 × 2 = 6). yes. - Decide. 51 = 3 × 17, so 51 has other factors. 51 is not prime.
Check. 3 × 17 = 51. ✓
Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.
- Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.We look for a factor that is not 1 or 91
- Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.Is it divisible by 2? The last digit of 91 is 1. Is it 0, 2, 4, 6 or 8? no.
Is it divisible by 3? 9 + 1 = 10. 10 is not in the 3 times table (9 and 12 are either side). no.
Is it divisible by 5? The last digit of 91 is 1. Is it 0 or 5? no.
Is it divisible by 7? 7 × 13 = 91 and 7 × 14 = 98. yes. - Decide. 91 = 7 × 13, so 91 has other factors. 91 is not prime.
Check. 7 × 13 = 91. ✓
Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.
- Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.We look for a factor that is not 1 or 97
- Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.Is it divisible by 2? The last digit of 97 is 7. Is it 0, 2, 4, 6 or 8? no.
Is it divisible by 3? 9 + 7 = 16. 16 is not in the 3 times table (15 and 18 are either side). no.
Is it divisible by 5? The last digit of 97 is 7. Is it 0 or 5? no.
Is it divisible by 7? 7 × 13 = 91 and 7 × 14 = 98. no. - Decide. None of 2, 3, 5 or 7 divides 97, so 97 is prime.
Check. The only factor pair of 97 is 1 × 97. ✓
Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Cross out the even numbers first. Then test each odd number that is left with the 3, 5 and 7 times tables.
- Write the numbers from 20 to 40. 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40
- Cross out the even numbers. Even numbers (except 2) are divisible by 2, so they are not prime.20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40
- Test each odd number that is left. Try the 3, 5 and 7 times tables. Stop when one divides it.21 = 3 × 7: not prime
23: not in the 3, 5 or 7 times tables: prime
25 = 5 × 5: not prime
27 = 3 × 9: not prime
29: not in the 3, 5 or 7 times tables: prime
31: not in the 3, 5 or 7 times tables: prime
33 = 3 × 11: not prime
35 = 5 × 7: not prime
37: not in the 3, 5 or 7 times tables: prime
39 = 3 × 13: not prime - Write the prime numbers. 23, 29, 31, 37
Check. Each prime has no factor except 1 and itself. 4 primes between 20 and 40. ✓
Watch out. Do not forget to test with 7. For example 91 = 7 × 13.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. The prime numbers are the gold squares on the finished sieve. Look along the rows for 61 to 79.
- Find the rows for the 60s and 70s. The 60s are the seventh row and the 70s are the eighth row.61, 62, 63, …, 69, 70 and 71, 72, 73, …, 79, 80
- Read the gold numbers. Gold numbers have not been crossed out, so they are prime.60s: 61, 67 70s: 71, 73, 79
- Write the answer. Check that none of them is 60 or 80: they are not in the list.61, 67, 71, 73, 79
Check. Test one with the divisibility tests: 79 is odd, 7 + 9 = 16 is not in the 3s, it does not end in 0 or 5, and 7 × 11 = 77 and 7 × 12 = 84, so it is not in the 7s. 79 is prime. ✓
Watch out. Between 60 and 80 does not include 60 or 80. The primes are never even (except 2) so you can skip every even number.
Watch out
Do not forget that 1 is not prime, and that 2 is prime (even though it is even). Also, numbers that are odd are not always prime: 9, 15, 21, 27 and 33 are odd but composite. And test with 7: 49, 77 and 91 are not in the 2, 3 or 5 times tables, but they are multiples of 7.Exercise 3E Prime and composite numbers
Non-calculatorKey words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find all the factors, count them, and decide.
- List the factors of 1. Use factor pairs.Factors: 1
- Count the factors. 1 factor
- Decide. Prime: exactly two factors. Composite: more than two. 1: only one.1 is neither. Only one factor, so not prime and not composite.
Check. Every number except 1 is either prime or composite. 1 has just 1 factor. ✓
Watch out. 1 is not prime. 2 is prime (it has exactly two factors).
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find all the factors, count them, and decide.
- List the factors of 7. Use factor pairs.1 × 7
Factors: 1, 7 - Count the factors. 2 factors
- Decide. Prime: exactly two factors. Composite: more than two. 1: only one.7 is prime. Exactly two factors, so prime.
Check. Every number except 1 is either prime or composite. 7 has 2 factors. ✓
Watch out. 1 is not prime. 2 is prime (it has exactly two factors).
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find all the factors, count them, and decide.
- List the factors of 9. Use factor pairs.1 × 9, 3 × 3
Factors: 1, 3, 9 - Count the factors. 3 factors
- Decide. Prime: exactly two factors. Composite: more than two. 1: only one.9 is composite. More than two factors, so composite.
Check. Every number except 1 is either prime or composite. 9 has 3 factors. ✓
Watch out. 1 is not prime. 2 is prime (it has exactly two factors).
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find all the factors, count them, and decide.
- List the factors of 11. Use factor pairs.1 × 11
Factors: 1, 11 - Count the factors. 2 factors
- Decide. Prime: exactly two factors. Composite: more than two. 1: only one.11 is prime. Exactly two factors, so prime.
Check. Every number except 1 is either prime or composite. 11 has 2 factors. ✓
Watch out. 1 is not prime. 2 is prime (it has exactly two factors).
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find all the factors, count them, and decide.
- List the factors of 15. Use factor pairs.1 × 15, 3 × 5
Factors: 1, 3, 5, 15 - Count the factors. 4 factors
- Decide. Prime: exactly two factors. Composite: more than two. 1: only one.15 is composite. More than two factors, so composite.
Check. Every number except 1 is either prime or composite. 15 has 4 factors. ✓
Watch out. 1 is not prime. 2 is prime (it has exactly two factors).
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find all the factors, count them, and decide.
- List the factors of 17. Use factor pairs.1 × 17
Factors: 1, 17 - Count the factors. 2 factors
- Decide. Prime: exactly two factors. Composite: more than two. 1: only one.17 is prime. Exactly two factors, so prime.
Check. Every number except 1 is either prime or composite. 17 has 2 factors. ✓
Watch out. 1 is not prime. 2 is prime (it has exactly two factors).
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test each number. A number is prime if its only factors are 1 and itself. If you can find any other factor, it is not prime.
- Test each number in turn. Look for a factor other than 1 and the number.2: factors 1, 2: prime
9 = 3 × 3: not prime
11: factors 1, 11: prime
15 = 3 × 5: not prime
17: factors 1, 17: prime
21 = 3 × 7: not prime
23: factors 1, 23: prime
27 = 3 × 9: not prime - Write the primes. 2, 11, 17, 23
Check. Each prime has exactly two factors. We found 4 primes. ✓
Watch out. 9, 15, 21 and 27 are odd but they are not prime.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Cross out the even numbers first. Then test each odd number that is left with the 3, 5 and 7 times tables.
- Write the numbers from 2 to 19. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
- Cross out the even numbers. Even numbers (except 2) are divisible by 2, so they are not prime.4, 6, 8, 10, 12, 14, 16, 18
- Test each odd number that is left. Try the 3, 5 and 7 times tables. Stop when one divides it.2: prime (the only even prime)
3: a prime we already know
5: a prime we already know
7: a prime we already know
9 = 3 × 3: not prime
11: not in the 3, 5 or 7 times tables: prime
13: not in the 3, 5 or 7 times tables: prime
15 = 3 × 5: not prime
17: not in the 3, 5 or 7 times tables: prime
19: not in the 3, 5 or 7 times tables: prime - Write the prime numbers. 2, 3, 5, 7, 11, 13, 17, 19
Check. Each prime has no factor except 1 and itself. 8 primes between 2 and 19. ✓
Watch out. Do not forget to test with 7. For example 91 = 7 × 13.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Count up through the primes in order: 2, 3, 5, 7, 11, 13, 17, 19. Find the ones that are missing from the list.
- Write the primes up to 19. You should know these.2, 3, 5, 7, 11, 13, 17, 19
- Compare with the list in the question. The list has 2, 3, __, 7, __, 13, __, 19.The gaps are between 3 and 7, between 7 and 13, and between 13 and 19.
- Fill the gaps. 5, 11, 17
Check. 5, 11 and 17 each have only two factors. ✓
Watch out. 9 and 15 are not prime, so they do not go in the gaps.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
| 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
| 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Look at the unshaded numbers. Test each one to see whether it is prime.
- Write the numbers that are not shaded. Do not include 2, 3 and 5 (gold) or 1 (purple).7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97
- Find the ones that are not prime. Test with 7: the multiples of 7 that are left.7 × 7 = 49 7 × 11 = 77 7 × 13 = 91
- Write the answer. All the others are prime.49, 77, 91
Check. 7 × 7 = 49, 7 × 11 = 77 and 7 × 13 = 91. ✓
Watch out. Do not include 1: it is neither prime nor composite.
Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…
Method. Look at the three numbers 49, 77 and 91.
- Divide each one by 7. 49 ÷ 7 = 7 77 ÷ 7 = 11 91 ÷ 7 = 13
- Say what they have in common. They all divide by 7, so they are in the 7 times table.They are all multiples of 7.
Check. These are the next step in the sieve: Maya would now shade the multiples of 7. ✓
Watch out. This is why the sieve (and the tests) stop at 7: after that nothing new is left to cross out up to 100.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.
- Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.We look for a factor that is not 1 or 39
- Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.Is it divisible by 2? The last digit of 39 is 9. Is it 0, 2, 4, 6 or 8? no.
Is it divisible by 3? 3 + 9 = 12. 12 is in the 3 times table (3 × 4 = 12). yes. - Decide. 39 = 3 × 13, so 39 has other factors. 39 is not prime.
Check. 3 × 13 = 39. ✓
Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.
- Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.We look for a factor that is not 1 or 53
- Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.Is it divisible by 2? The last digit of 53 is 3. Is it 0, 2, 4, 6 or 8? no.
Is it divisible by 3? 5 + 3 = 8. 8 is not in the 3 times table (6 and 9 are either side). no.
Is it divisible by 5? The last digit of 53 is 3. Is it 0 or 5? no.
Is it divisible by 7? 7 × 7 = 49 and 7 × 8 = 56. no. - Decide. None of 2, 3, 5 or 7 divides 53, so 53 is prime.
Check. The only factor pair of 53 is 1 × 53. ✓
Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.
- Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.We look for a factor that is not 1 or 57
- Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.Is it divisible by 2? The last digit of 57 is 7. Is it 0, 2, 4, 6 or 8? no.
Is it divisible by 3? 5 + 7 = 12. 12 is in the 3 times table (3 × 4 = 12). yes. - Decide. 57 = 3 × 19, so 57 has other factors. 57 is not prime.
Check. 3 × 19 = 57. ✓
Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.
- Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.We look for a factor that is not 1 or 71
- Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.Is it divisible by 2? The last digit of 71 is 1. Is it 0, 2, 4, 6 or 8? no.
Is it divisible by 3? 7 + 1 = 8. 8 is not in the 3 times table (6 and 9 are either side). no.
Is it divisible by 5? The last digit of 71 is 1. Is it 0 or 5? no.
Is it divisible by 7? 7 × 10 = 70 and 7 × 11 = 77. no. - Decide. None of 2, 3, 5 or 7 divides 71, so 71 is prime.
Check. The only factor pair of 71 is 1 × 71. ✓
Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.
- Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.We look for a factor that is not 1 or 87
- Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.Is it divisible by 2? The last digit of 87 is 7. Is it 0, 2, 4, 6 or 8? no.
Is it divisible by 3? 8 + 7 = 15. 15 is in the 3 times table (3 × 5 = 15). yes. - Decide. 87 = 3 × 29, so 87 has other factors. 87 is not prime.
Check. 3 × 29 = 87. ✓
Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.
- Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.We look for a factor that is not 1 or 89
- Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.Is it divisible by 2? The last digit of 89 is 9. Is it 0, 2, 4, 6 or 8? no.
Is it divisible by 3? 8 + 9 = 17. 17 is not in the 3 times table (15 and 18 are either side). no.
Is it divisible by 5? The last digit of 89 is 9. Is it 0 or 5? no.
Is it divisible by 7? 7 × 12 = 84 and 7 × 13 = 91. no. - Decide. None of 2, 3, 5 or 7 divides 89, so 89 is prime.
Check. The only factor pair of 89 is 1 × 89. ✓
Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Cross out the even numbers first. Then test each odd number that is left with the 3, 5 and 7 times tables.
- Write the numbers from 30 to 50. 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50
- Cross out the even numbers. Even numbers (except 2) are divisible by 2, so they are not prime.30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50
- Test each odd number that is left. Try the 3, 5 and 7 times tables. Stop when one divides it.31: not in the 3, 5 or 7 times tables: prime
33 = 3 × 11: not prime
35 = 5 × 7: not prime
37: not in the 3, 5 or 7 times tables: prime
39 = 3 × 13: not prime
41: not in the 3, 5 or 7 times tables: prime
43: not in the 3, 5 or 7 times tables: prime
45 = 3 × 15: not prime
47: not in the 3, 5 or 7 times tables: prime
49 = 7 × 7: not prime - Write the prime numbers. 31, 37, 41, 43, 47
Check. Each prime has no factor except 1 and itself. 5 primes between 30 and 50. ✓
Watch out. Do not forget to test with 7. For example 91 = 7 × 13.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find a prime number that divides 35. Then check that the other number is prime too.
- Try the small primes in order. Use 2, 3, 5, 7 and the tests of divisibility.35 is divisible by 5: 35 ÷ 5 = 7
- Check that 7 is prime. Both numbers must be prime.7 has factors 1, 7, so it is prime.
- Write the answer. 5 × 7 = 35, so the primes are 5 and 7
Check. 5 × 7 = 35. ✓
Watch out. Both numbers must be prime. For example 6 × 6 = 36 does not count, because 6 is not prime.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find a prime number that divides 77. Then check that the other number is prime too.
- Try the small primes in order. Use 2, 3, 5, 7 and the tests of divisibility.77 is divisible by 7: 77 ÷ 7 = 11
- Check that 11 is prime. Both numbers must be prime.11 has factors 1, 11, so it is prime.
- Write the answer. 7 × 11 = 77, so the primes are 7 and 11
Check. 7 × 11 = 77. ✓
Watch out. Both numbers must be prime. For example 6 × 6 = 36 does not count, because 6 is not prime.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Find a prime number that divides 91. Then check that the other number is prime too.
- Try the small primes in order. Use 2, 3, 5, 7 and the tests of divisibility.91 is divisible by 7: 91 ÷ 7 = 13
- Check that 13 is prime. Both numbers must be prime.13 has factors 1, 13, so it is prime.
- Write the answer. 7 × 13 = 91, so the primes are 7 and 13
Check. 7 × 13 = 91. ✓
Watch out. Both numbers must be prime. For example 6 × 6 = 36 does not count, because 6 is not prime.
Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.
Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.
- Start with the clue that gives a short list: “a prime number”. List the numbers between 30 and 50 that fit this clue.31, 37, 41, 43, 47
- Use the next clue: “its digits add up to 7”. Cross out every number that does not fit.
31,37,41, 43,47 - Check the answer against every clue. It must fit all of them.“a prime number” ✓
“its digits add up to 7” ✓
Check. Only 43 is left, and it fits every clue. ✓
Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Test some odd numbers. If one is not prime, Mia is wrong.
- Try 3, 5, 7. These are odd.Each has only two factors, so 3, 5 and 7 are prime.
- Try 9. 9 is odd.9 = 3 × 3, so the factors of 9 are 1, 3, 9. That is three factors, so 9 is not prime.
- Answer Mia. One example that breaks the rule is enough.No. 9 is odd but not prime. (15 and 21 are also odd and composite.)
Check. 9 ÷ 3 = 3 exactly, so 3 is a factor of 9 other than 1 and 9. ✓
Watch out. Odd does not mean prime. Test for factors, not just for odd or even.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.3, 5, 7 and 11 are prime and odd.
- Look for one that does not work. One example that breaks the rule is called a counter-example.2 is prime but it is even.
- Explain why. A reason shows what is happening for every number, not just the ones we tried.Every even number except 2 has 2 as a factor, so it cannot be prime. So all primes except 2 are odd, but 2 is the only even prime.
- Decide. Sometimes
Check. 2 has factors 1 and 2, so it is prime. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.6 has factors 1, 2, 3, 6 (4 factors): not prime.
7 has factors 1, 7 (2 factors): prime.
8 has factors 1, 2, 4, 8 (4 factors): not prime.
13 has factors 1 and 13 (2 factors): prime. - Explain why. A reason shows what is happening for every number, not just the ones we tried.The definition of a prime number is a number with exactly two factors (1 and itself). So a number with exactly two factors is a prime number.
- Decide. Always
Check. 11 has exactly two factors (1 and 11) and it is prime. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A counter-example is one example that shows a statement is false.
Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.
- Try some numbers. Test the statement with some examples.2 + 3 = 5, and 5 is prime. ✓
- Look for one that does not work. One example that breaks the rule is called a counter-example.3 + 5 = 8, and 8 is not prime (it is even and bigger than 2).
- Explain why. A reason shows what is happening for every number, not just the ones we tried.Two odd primes add to an even number bigger than 2, which is never prime. But 2 + an odd prime makes an odd number, which can be prime (2 + 3 = 5, 2 + 5 = 7, 2 + 11 = 13).
- Decide. Sometimes
Check. 5 is prime and 8 is not. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Look for something that three of the numbers have in common. Try listing the factors.
- List the factors of each number. Count them.11: 1, 11
13: 1, 13
17: 1, 17
21: 1, 3, 7, 21 - Compare. 11, 13 and 17 have exactly two factors. 21 has four.11, 13 and 17 are prime. 21 is composite.
- Choose the odd one out. 21 (it is not prime: 21 = 3 × 7)
Check. 3 × 7 = 21, so 21 has the factors 3 and 7 as well as 1 and 21. ✓
Watch out. All four numbers are odd, so “odd” is not the reason.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Look at the factors of some even numbers. Then explain why it always happens.
- Try some even numbers bigger than 2. List the factors.4: 1, 2, 4
6: 1, 2, 3, 6
10: 1, 2, 5, 10
18: 1, 2, 3, 6, 9, 18 - Spot the pattern. Every list has the number 2 in it.An even number divides exactly by 2.
- Explain. An even number bigger than 2 has the factors 1, 2 and itself. That is at least three different factors. A prime number has exactly two.So it cannot be prime.
Check. The only even prime is 2, with factors 1 and 2. ✓
Watch out. A reason is needed, not just examples.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Work out the correct answer yourself first. Then compare it with the answer in the question to find the mistake.
- Work it out carefully. Do the question again, one step at a time.The last digit of 51 is 1. Is it 0, 2, 4, 6 or 8? no.
5 + 1 = 6. 6 is in the 3 times table (3 × 2 = 6). yes.
51 = 3 × 17 - Compare with the answer in the question. Amira said 51 is prime.
- Find the mistake. Amira only tested 2, 5 and 10. She forgot to test 3. The digits of 51 add up to 6, which is in the 3 times table.
- Write the correct answer. 51 is not prime. 51 = 3 × 17.
Check. 3 × 17 = 51. ✓
Watch out. Test with 2, 3, 5 and 7. Do not skip 3.
Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.
Method. Start with each prime that is less than 15. Take it away from 30 to find the partner. Then check that the partner is prime too.
- List the primes up to 15. One number in each pair is 15 or less.2, 3, 5, 7, 11, 13
- Find the partner of each. Take the prime away from 30. Is the answer prime?
Prime 30 minus the prime Is it prime? 2 28 no 3 27 no 5 25 no 7 23 yes 11 19 yes 13 17 yes - Write the pairs that work. 7 + 23, 11 + 19, 13 + 17
Check. 7 + 23 = 30, 11 + 19 = 30 and 13 + 17 = 30, and all six numbers are prime. ✓
Watch out. Both numbers must be prime. 2 + 28 and 3 + 27 do not work.