GCSE Maths: Prime numbers (Higher)
Part of the GCSE Maths factors, multiples and primes chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Prime numbers
Some numbers have lots of factors. 12 has six: 1, 2, 3, 4, 6 and 12. Other numbers have very few. The number 7 can only be made by 1 × 7, so its only factors are 1 and 7. A number like this is called a prime number.
A prime number is a whole number that has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19 ... A number with more than two factors, such as 12, is called a composite number.
Two numbers cause trouble. 1 is not prime, because it has only one factor (1 itself). 2 is prime, and it is the only even prime, because every other even number is also divisible by 2, giving it a third factor.
To find the primes this way, shade 2 then cross out all its multiples, move to the next number not crossed out (3), shade it and cross out its multiples, and so on. Any number left must be prime. You should learn the primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
How can you test a larger number such as 91 without writing out all its factors? Factors come in pairs, one small and one large. If a number has a factor bigger than its square root, its partner is smaller than the square root. So we only need to test primes up to the square root. For 91, 9 × 9 = 81 and 10 × 10 = 100, so the square root is between 9 and 10. We test 2, 3, 5 and 7. Since 91 = 7 × 13, it is not prime.
Key ideas
- A prime has exactly two factors: 1 and itself.
- 1 is not prime. 2 is the only even prime.
- A number with more than two factors is composite.
- Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
- To test a number, divide by the primes up to its square root. If none goes in exactly, it is prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √29 divides into 29.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 29 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 29 had a factor bigger than √29, its partner would be smaller than √29 and we would meet it first. Here 5 × 5 = 25 < 29 and 6 × 6 = 36 > 29, so we only need to test the primes up to 5. We test primes only, because any other divisor is built from primes.Primes up to 5: 2, 3, 5
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.29 ÷ 2 = 14 remainder 1, so 2 is not a factor.
29 ÷ 3 = 9 remainder 2, so 3 is not a factor.
29 ÷ 5 = 5 remainder 4, so 5 is not a factor. - Decide. None of the primes up to 5 divides into 29, so 29 has no factors except 1 and itself.29 is prime.
Check. Its only factors are 1 and 29. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √91 divides into 91.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 91 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 91 had a factor bigger than √91, its partner would be smaller than √91 and we would meet it first. Here 9 × 9 = 81 < 91 and 10 × 10 = 100 > 91, so we only need to test the primes up to 9. We test primes only, because any other divisor is built from primes.Primes up to 9: 2, 3, 5, 7
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.91 ÷ 2 = 45 remainder 1, so 2 is not a factor.
91 ÷ 3 = 30 remainder 1, so 3 is not a factor.
91 ÷ 5 = 18 remainder 1, so 5 is not a factor.
91 ÷ 7 = 13 exactly, so 7 is a factor. - Decide. 7 is a factor other than 1 and 91, so there are more than two factors.91 = 7 × 13, so 91 is not prime.
Check. 7 × 13 = 91. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Check each number from 40 to 60: if a small prime divides it, it is not prime.
- List the numbers to check. Every number in the range must be tested.40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60
- Test each one. Look for a factor other than 1 and the number itself (try 2, 3, 5, 7 ...).40 = 2 × 20, not prime
41: no prime up to √41 divides it, so prime ✓
42 = 2 × 21, not prime
43: no prime up to √43 divides it, so prime ✓
44 = 2 × 22, not prime
45 = 3 × 15, not prime
46 = 2 × 23, not prime
47: no prime up to √47 divides it, so prime ✓
48 = 2 × 24, not prime
49 = 7 × 7, not prime
50 = 2 × 25, not prime
51 = 3 × 17, not prime
52 = 2 × 26, not prime
53: no prime up to √53 divides it, so prime ✓
54 = 2 × 27, not prime
55 = 5 × 11, not prime
56 = 2 × 28, not prime
57 = 3 × 19, not prime
58 = 2 × 29, not prime
59: no prime up to √59 divides it, so prime ✓
60 = 2 × 30, not prime - Write the primes. 41, 43, 47, 53, 59
Check. Count: 5 primes in the range. ✓
Watch out. Even numbers bigger than 2 are never prime, and numbers ending in 5 (bigger than 5) are never prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Look for a common factor. If the whole expression has a factor other than 1 and itself, it cannot be prime.
- Look at the two parts of the sum. There are two terms: 3 × 5 × 7 and 7. Both contain a 7.3 × 5 × 7 + 7
- Take the common factor out. Both terms are multiples of 7, so we can write the sum as 7 times a number.3 × 5 × 7 + 7 = 7 × (3 × 5 + 1) = 7 × 16
- Explain. The number has a factor of 7, which is neither 1 nor the number itself.It is a multiple of 7 that is bigger than 7, so it is not prime.
Check. 3 × 5 × 7 + 7 = 105 + 7 = 112, and 112 = 7 × 16. ✓
Watch out. “Explain” needs a sentence naming the factor, not just the value.
Watch out
Do not forget that 1 is not prime and 2 is prime. Also, a number is not prime just because it is odd: 9, 15, 21, 25, 27 and 91 are odd but have other factors. And 91 looks prime but it is 7 × 13.Exercise 2B Prime numbers
Non-calculatorKey words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. A prime number has exactly two factors, 1 and itself. Test each number by looking for another factor.
- Test each number. Look for a factor other than 1 and the number itself.2: no number except 1 and 2 divides it, so prime ✓
9 = 3 × 3, so not prime
11: no number except 1 and 11 divides it, so prime ✓
15 = 3 × 5, so not prime
17: no number except 1 and 17 divides it, so prime ✓ - Write down the primes. Keep only the ones with no other factor.2, 11, 17
Check. Each prime in the answer has no factor except 1 and itself. ✓
Watch out. 1 is not a prime number, and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. A prime number has exactly two factors, 1 and itself. Test each number by looking for another factor.
- Test each number. Look for a factor other than 1 and the number itself.1 has only one factor, so it is not prime.
4 = 2 × 2, so not prime
13: no number except 1 and 13 divides it, so prime ✓
21 = 3 × 7, so not prime
23: no number except 1 and 23 divides it, so prime ✓ - Write down the primes. Keep only the ones with no other factor.13, 23
Check. Each prime in the answer has no factor except 1 and itself. ✓
Watch out. 1 is not a prime number, and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. A prime number has exactly two factors, 1 and itself. Test each number by looking for another factor.
- Test each number. Look for a factor other than 1 and the number itself.27 = 3 × 9, so not prime
29: no number except 1 and 29 divides it, so prime ✓
31: no number except 1 and 31 divides it, so prime ✓
33 = 3 × 11, so not prime - Write down the primes. Keep only the ones with no other factor.29, 31
Check. Each prime in the answer has no factor except 1 and itself. ✓
Watch out. 1 is not a prime number, and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. A prime number has exactly two factors, 1 and itself. Test each number by looking for another factor.
- Test each number. Look for a factor other than 1 and the number itself.35 = 5 × 7, so not prime
37: no number except 1 and 37 divides it, so prime ✓
39 = 3 × 13, so not prime
41: no number except 1 and 41 divides it, so prime ✓ - Write down the primes. Keep only the ones with no other factor.37, 41
Check. Each prime in the answer has no factor except 1 and itself. ✓
Watch out. 1 is not a prime number, and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. A prime number has exactly two factors, 1 and itself. Test each number by looking for another factor.
- Test each number. Look for a factor other than 1 and the number itself.43: no number except 1 and 43 divides it, so prime ✓
45 = 3 × 15, so not prime
47: no number except 1 and 47 divides it, so prime ✓
49 = 7 × 7, so not prime - Write down the primes. Keep only the ones with no other factor.43, 47
Check. Each prime in the answer has no factor except 1 and itself. ✓
Watch out. 1 is not a prime number, and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. A prime number has exactly two factors, 1 and itself. Test each number by looking for another factor.
- Test each number. Look for a factor other than 1 and the number itself.51 = 3 × 17, so not prime
53: no number except 1 and 53 divides it, so prime ✓
55 = 5 × 11, so not prime
57 = 3 × 19, so not prime - Write down the primes. Keep only the ones with no other factor.53
Check. Each prime in the answer has no factor except 1 and itself. ✓
Watch out. 1 is not a prime number, and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Check each number from 10 to 30: if a small prime divides it, it is not prime.
- List the numbers to check. Every number in the range must be tested.10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30
- Test each one. Look for a factor other than 1 and the number itself (try 2, 3, 5, 7 ...).10 = 2 × 5, not prime
11: no prime up to √11 divides it, so prime ✓
12 = 2 × 6, not prime
13: no prime up to √13 divides it, so prime ✓
14 = 2 × 7, not prime
15 = 3 × 5, not prime
16 = 2 × 8, not prime
17: no prime up to √17 divides it, so prime ✓
18 = 2 × 9, not prime
19: no prime up to √19 divides it, so prime ✓
20 = 2 × 10, not prime
21 = 3 × 7, not prime
22 = 2 × 11, not prime
23: no prime up to √23 divides it, so prime ✓
24 = 2 × 12, not prime
25 = 5 × 5, not prime
26 = 2 × 13, not prime
27 = 3 × 9, not prime
28 = 2 × 14, not prime
29: no prime up to √29 divides it, so prime ✓
30 = 2 × 15, not prime - Write the primes. 11, 13, 17, 19, 23, 29
Check. Count: 6 primes in the range. ✓
Watch out. Even numbers bigger than 2 are never prime, and numbers ending in 5 (bigger than 5) are never prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √51 divides into 51.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 51 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 51 had a factor bigger than √51, its partner would be smaller than √51 and we would meet it first. Here 7 × 7 = 49 < 51 and 8 × 8 = 64 > 51, so we only need to test the primes up to 7. We test primes only, because any other divisor is built from primes.Primes up to 7: 2, 3, 5, 7
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.51 ÷ 2 = 25 remainder 1, so 2 is not a factor.
51 ÷ 3 = 17 exactly, so 3 is a factor. - Decide. 3 is a factor other than 1 and 51, so there are more than two factors.51 = 3 × 17, so 51 is not prime.
Check. 3 × 17 = 51. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √57 divides into 57.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 57 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 57 had a factor bigger than √57, its partner would be smaller than √57 and we would meet it first. Here 7 × 7 = 49 < 57 and 8 × 8 = 64 > 57, so we only need to test the primes up to 7. We test primes only, because any other divisor is built from primes.Primes up to 7: 2, 3, 5, 7
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.57 ÷ 2 = 28 remainder 1, so 2 is not a factor.
57 ÷ 3 = 19 exactly, so 3 is a factor. - Decide. 3 is a factor other than 1 and 57, so there are more than two factors.57 = 3 × 19, so 57 is not prime.
Check. 3 × 19 = 57. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √61 divides into 61.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 61 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 61 had a factor bigger than √61, its partner would be smaller than √61 and we would meet it first. Here 7 × 7 = 49 < 61 and 8 × 8 = 64 > 61, so we only need to test the primes up to 7. We test primes only, because any other divisor is built from primes.Primes up to 7: 2, 3, 5, 7
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.61 ÷ 2 = 30 remainder 1, so 2 is not a factor.
61 ÷ 3 = 20 remainder 1, so 3 is not a factor.
61 ÷ 5 = 12 remainder 1, so 5 is not a factor.
61 ÷ 7 = 8 remainder 5, so 7 is not a factor. - Decide. None of the primes up to 7 divides into 61, so 61 has no factors except 1 and itself.61 is prime.
Check. Its only factors are 1 and 61. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √77 divides into 77.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 77 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 77 had a factor bigger than √77, its partner would be smaller than √77 and we would meet it first. Here 8 × 8 = 64 < 77 and 9 × 9 = 81 > 77, so we only need to test the primes up to 8. We test primes only, because any other divisor is built from primes.Primes up to 8: 2, 3, 5, 7
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.77 ÷ 2 = 38 remainder 1, so 2 is not a factor.
77 ÷ 3 = 25 remainder 2, so 3 is not a factor.
77 ÷ 5 = 15 remainder 2, so 5 is not a factor.
77 ÷ 7 = 11 exactly, so 7 is a factor. - Decide. 7 is a factor other than 1 and 77, so there are more than two factors.77 = 7 × 11, so 77 is not prime.
Check. 7 × 11 = 77. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √87 divides into 87.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 87 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 87 had a factor bigger than √87, its partner would be smaller than √87 and we would meet it first. Here 9 × 9 = 81 < 87 and 10 × 10 = 100 > 87, so we only need to test the primes up to 9. We test primes only, because any other divisor is built from primes.Primes up to 9: 2, 3, 5, 7
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.87 ÷ 2 = 43 remainder 1, so 2 is not a factor.
87 ÷ 3 = 29 exactly, so 3 is a factor. - Decide. 3 is a factor other than 1 and 87, so there are more than two factors.87 = 3 × 29, so 87 is not prime.
Check. 3 × 29 = 87. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √89 divides into 89.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 89 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 89 had a factor bigger than √89, its partner would be smaller than √89 and we would meet it first. Here 9 × 9 = 81 < 89 and 10 × 10 = 100 > 89, so we only need to test the primes up to 9. We test primes only, because any other divisor is built from primes.Primes up to 9: 2, 3, 5, 7
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.89 ÷ 2 = 44 remainder 1, so 2 is not a factor.
89 ÷ 3 = 29 remainder 2, so 3 is not a factor.
89 ÷ 5 = 17 remainder 4, so 5 is not a factor.
89 ÷ 7 = 12 remainder 5, so 7 is not a factor. - Decide. None of the primes up to 9 divides into 89, so 89 has no factors except 1 and itself.89 is prime.
Check. Its only factors are 1 and 89. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √101 divides into 101.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 101 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 101 had a factor bigger than √101, its partner would be smaller than √101 and we would meet it first. Here 10 × 10 = 100 < 101 and 11 × 11 = 121 > 101, so we only need to test the primes up to 10. We test primes only, because any other divisor is built from primes.Primes up to 10: 2, 3, 5, 7
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.101 ÷ 2 = 50 remainder 1, so 2 is not a factor.
101 ÷ 3 = 33 remainder 2, so 3 is not a factor.
101 ÷ 5 = 20 remainder 1, so 5 is not a factor.
101 ÷ 7 = 14 remainder 3, so 7 is not a factor. - Decide. None of the primes up to 10 divides into 101, so 101 has no factors except 1 and itself.101 is prime.
Check. Its only factors are 1 and 101. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √119 divides into 119.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 119 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 119 had a factor bigger than √119, its partner would be smaller than √119 and we would meet it first. Here 10 × 10 = 100 < 119 and 11 × 11 = 121 > 119, so we only need to test the primes up to 10. We test primes only, because any other divisor is built from primes.Primes up to 10: 2, 3, 5, 7
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.119 ÷ 2 = 59 remainder 1, so 2 is not a factor.
119 ÷ 3 = 39 remainder 2, so 3 is not a factor.
119 ÷ 5 = 23 remainder 4, so 5 is not a factor.
119 ÷ 7 = 17 exactly, so 7 is a factor. - Decide. 7 is a factor other than 1 and 119, so there are more than two factors.119 = 7 × 17, so 119 is not prime.
Check. 7 × 17 = 119. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √133 divides into 133.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 133 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 133 had a factor bigger than √133, its partner would be smaller than √133 and we would meet it first. Here 11 × 11 = 121 < 133 and 12 × 12 = 144 > 133, so we only need to test the primes up to 11. We test primes only, because any other divisor is built from primes.Primes up to 11: 2, 3, 5, 7, 11
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.133 ÷ 2 = 66 remainder 1, so 2 is not a factor.
133 ÷ 3 = 44 remainder 1, so 3 is not a factor.
133 ÷ 5 = 26 remainder 3, so 5 is not a factor.
133 ÷ 7 = 19 exactly, so 7 is a factor. - Decide. 7 is a factor other than 1 and 133, so there are more than two factors.133 = 7 × 19, so 133 is not prime.
Check. 7 × 19 = 133. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √143 divides into 143.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 143 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 143 had a factor bigger than √143, its partner would be smaller than √143 and we would meet it first. Here 11 × 11 = 121 < 143 and 12 × 12 = 144 > 143, so we only need to test the primes up to 11. We test primes only, because any other divisor is built from primes.Primes up to 11: 2, 3, 5, 7, 11
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.143 ÷ 2 = 71 remainder 1, so 2 is not a factor.
143 ÷ 3 = 47 remainder 2, so 3 is not a factor.
143 ÷ 5 = 28 remainder 3, so 5 is not a factor.
143 ÷ 7 = 20 remainder 3, so 7 is not a factor.
143 ÷ 11 = 13 exactly, so 11 is a factor. - Decide. 11 is a factor other than 1 and 143, so there are more than two factors.143 = 11 × 13, so 143 is not prime.
Check. 11 × 13 = 143. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √149 divides into 149.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 149 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 149 had a factor bigger than √149, its partner would be smaller than √149 and we would meet it first. Here 12 × 12 = 144 < 149 and 13 × 13 = 169 > 149, so we only need to test the primes up to 12. We test primes only, because any other divisor is built from primes.Primes up to 12: 2, 3, 5, 7, 11
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.149 ÷ 2 = 74 remainder 1, so 2 is not a factor.
149 ÷ 3 = 49 remainder 2, so 3 is not a factor.
149 ÷ 5 = 29 remainder 4, so 5 is not a factor.
149 ÷ 7 = 21 remainder 2, so 7 is not a factor.
149 ÷ 11 = 13 remainder 6, so 11 is not a factor. - Decide. None of the primes up to 12 divides into 149, so 149 has no factors except 1 and itself.149 is prime.
Check. Its only factors are 1 and 149. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Test whether any prime up to √157 divides into 157.
- Remember what prime means. A prime has exactly two factors: 1 and itself. So if any other number divides into it, it is not prime.To show 157 is not prime we need one extra factor.
- Decide which numbers to test. Factors come in pairs. If 157 had a factor bigger than √157, its partner would be smaller than √157 and we would meet it first. Here 12 × 12 = 144 < 157 and 13 × 13 = 169 > 157, so we only need to test the primes up to 12. We test primes only, because any other divisor is built from primes.Primes up to 12: 2, 3, 5, 7, 11
- Divide by each of these primes in turn. Stop if one goes in exactly. One exact division proves that the number is not prime.157 ÷ 2 = 78 remainder 1, so 2 is not a factor.
157 ÷ 3 = 52 remainder 1, so 3 is not a factor.
157 ÷ 5 = 31 remainder 2, so 5 is not a factor.
157 ÷ 7 = 22 remainder 3, so 7 is not a factor.
157 ÷ 11 = 14 remainder 3, so 11 is not a factor. - Decide. None of the primes up to 12 divides into 157, so 157 has no factors except 1 and itself.157 is prime.
Check. Its only factors are 1 and 157. ✓
Watch out. You only need to test primes, and only up to the square root. Do not forget that 2 is prime (the only even prime).
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Use the facts about primes to find the number: the only even prime.
- Write down the relevant numbers. Collect the numbers that fit the description.Every even number is divisible by 2.
- Test them for primes. A prime has exactly two factors.An even number bigger than 2 has the factors 1, 2 and itself, so it is not prime. 2 has only the factors 1 and 2.
- Answer. Choose the prime that fits.2
Check. 2 has exactly two factors, 1 and 2. ✓
Watch out. Remember 1 is not prime and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Use the facts about primes to find the number: between 90 and 100.
- Write down the relevant numbers. Collect the numbers that fit the description.91 = 7 × 13, 92 even, 93 = 3 × 31, 94 even, 95 ends in 5, 96 even, 97 ?, 98 even, 99 = 9 × 11
- Test them for primes. A prime has exactly two factors.Test 97 with 2, 3, 5, 7: 97 ÷ 2 = 48 r 1, ÷ 3 = 32 r 1, ÷ 5 = 19 r 2, ÷ 7 = 13 r 6. None goes in.
- Answer. Choose the prime that fits.97
Check. 97 has no prime factor up to 9. ✓
Watch out. Remember 1 is not prime and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Use the facts about primes to find the number: one more than 4 squared.
- Write down the relevant numbers. Collect the numbers that fit the description.42 = 16 and 16 + 1 = 17
- Test them for primes. A prime has exactly two factors.17 ÷ 2 = 8 r 1, ÷ 3 = 5 r 2. 5 × 5 = 25 is more than 17, so stop. None goes in.
- Answer. Choose the prime that fits.17
Check. 17 has only the factors 1 and 17. ✓
Watch out. Remember 1 is not prime and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Use the facts about primes to find the number: the smallest prime above 50.
- Write down the relevant numbers. Collect the numbers that fit the description.51 = 3 × 17, 52 even, 53 ?
- Test them for primes. A prime has exactly two factors.53 ÷ 2 = 26 r 1, ÷ 3 = 17 r 2, ÷ 5 = 10 r 3, ÷ 7 = 7 r 4. None goes in.
- Answer. Choose the prime that fits.53
Check. 51 and 52 are not prime and 53 is. ✓
Watch out. Remember 1 is not prime and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Use the facts about primes to find the number: the largest prime below 30.
- Write down the relevant numbers. Collect the numbers that fit the description.29, 28 (even), 27 = 3 × 9
- Test them for primes. A prime has exactly two factors.29 ÷ 2 = 14 r 1, ÷ 3 = 9 r 2, ÷ 5 = 5 r 4. None goes in.
- Answer. Choose the prime that fits.29
Check. 29 is prime and 30 is not below 30. ✓
Watch out. Remember 1 is not prime and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Use the facts about primes to find the number: a factor of 35 that is prime and bigger than 5.
- Write down the relevant numbers. Collect the numbers that fit the description.Factors of 35: 1, 5, 7, 35
- Test them for primes. A prime has exactly two factors.7 is prime (1 × 7 only). 35 = 5 × 7 is not prime. 1 and 5 are not bigger than 5.
- Answer. Choose the prime that fits.7
Check. 35 ÷ 7 = 5. ✓
Watch out. Remember 1 is not prime and 2 is the only even prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. List the primes up to 30 and count them.
- Go through the numbers 1 to 30. Cross out 1, then all the multiples of 2, 3 and 5 (apart from 2, 3, 5 themselves). Larger primes would need to be tested, but 7 × 7 = 49 is more than 30.Crossed out: 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30
- Write what is left. These are the prime numbers.2, 3, 5, 7, 11, 13, 17, 19, 23, 29
- Count them. Count the numbers in the list.10 winning tickets
Check. Count the primes by tens: 4 below 10 (2, 3, 5, 7), 4 from 10 to 19 (11, 13, 17, 19) and 2 from 20 to 29 (23, 29): 4 + 4 + 2 = 10. ✓
Watch out. Do not count 1 as a prime.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Find the factor pair of 77 where both numbers are prime.
- Test the small primes. 77 is odd, so not 2. Digits 7 + 7 = 14, so not 3. It does not end in 0 or 5. Try 7.77 ÷ 7 = 11
- Check both are prime. 7 and 11 each have only two factors.7 = 7 × 1 and 11 = 11 × 1
- Add them. 7 + 11 = 18
Check. 7 × 11 = 77. ✓
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Find the factor pair of 143 where both numbers are prime.
- Test the small primes. 143: not even, digit sum 8 so not 3, not ending in 0 or 5, and 143 ÷ 7 = 20 r 3. Try 11.143 ÷ 11 = 13
- Check both are prime. 11 and 13 each have only two factors.11 and 13 are both on the list of primes
- Add them. 11 + 13 = 24
Check. 11 × 13 = 143. ✓
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Use the definition: a prime has exactly two factors.
- Take any even number bigger than 2. Even numbers are multiples of 2.For example 10, 16, 40
- List what we know divides it. 1, 2 and the number itself all divide it exactly, and these are three different numbers.10: 1, 2, 5, 10 (at least 1, 2 and 10)
- Conclude. More than two factors means not prime. The number 2 itself has only the factors 1 and 2.So 2 is the only even prime.
Check. 4, 6, 8, 10, 12 each have 1, 2 and themselves as factors. ✓
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Find one odd number that is not prime (a counter-example).
- Test the odd numbers in order. 1 is not prime; 3, 5 and 7 are prime. Try 9.9 ÷ 3 = 3 exactly
- Write it as a product. This gives a factor other than 1 and 9.9 = 3 × 3
- Conclude. One counter-example is enough.9 is odd but it has three factors (1, 3, 9), so Zoe is wrong.
Check. 15 = 3 × 5 and 21 = 3 × 7 are also odd non-primes. ✓
Watch out. A counter-example is one specific example that breaks the statement.
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Compare 1 with the definition: exactly two factors.
- Count the factors of 1. Test which numbers divide 1.1 ÷ 1 = 1, so the only factor of 1 is 1
- Compare with the definition. A prime has exactly two different factors.1 has one factor, not two
- Conclude. Sam is wrong: 1 is not prime.
Check. The smallest prime is 2, which has the two factors 1 and 2. ✓
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Part (a) is a multiplication. For (b) show that the six-digit number is the three-digit number multiplied by 1001.
- (a) Multiply 7 × 11 first. Multiply two numbers at a time.7 × 11 = 77
- (a) Multiply the result by 13. Use long multiplication: 77 × 10 = 770 and 77 × 3 = 231.770 + 231 = 1001
- (b) Write 247 247 as a sum. 247 247 is 247 thousand plus 247, so it is 247 × 1000 + 247.247 247 = 247 × 1000 + 247 × 1 = 247 × 1001
- (b) Explain. The same works for any three-digit number abc: abcabc = abc × 1001. So it has a factor of 1001 (and of 7, 11 and 13), which is not 1 or itself.So a number of this form always has more than two factors, so it is never prime.
Check. 247 × 1001 = 247 247. ✓
Key words. A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11 ...). A composite number has more than two factors. 1 is neither prime nor composite because it has only one factor.
Method. Use the product to find the possible pairs, then use the sum to choose.
- Find the factor pairs of 143. Test small primes: 143 ÷ 11 = 13.143 = 11 × 13 (and 1 × 143)
- Check that the numbers are prime. Both 11 and 13 have only two factors.11 and 13 are prime. 143 itself is not.
- Check the sum. 11 + 13 = 24 ✓
Check. 11 × 13 = 143. ✓
Watch out. 1 × 143 is not allowed because 1 is not prime.