Year 6: Prime numbers up to 100
Part of the Year 6 hcf, lcm and prime numbers chapter. Chapter overview · whole chapter on one page
Prime numbers up to 100
A prime number has exactly two factors: 1 and itself. The number 7 is prime, because its only factors are 1 and 7. 1 is not prime, because it has only one factor. 2 is the only even prime, because every other even number has 2 as a factor.
A number that is not prime is called composite. 9 is composite, because 9 = 3 × 3. Its factors are 1, 3 and 9.
| 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 |
To find the primes up to 100, cross out the multiples of 2, 3, 5 and 7 (except the numbers themselves). The numbers left are prime. We only need the primes up to 7, because 11 × 11 = 121 is more than 100.
To check a number is prime, divide it by each prime up to its square root. If none of them divides it exactly, the number is prime.
Key ideas
- A prime number has exactly two factors: 1 and itself. 1 is not prime.
- 2 is the only even prime number. Every other even number has a factor of 2.
- To test a number up to 100, try the primes up to its square root: 2, 3, 5 and 7 are enough.
- The primes up to 20 are 2, 3, 5, 7, 11, 13, 17 and 19. Learn them.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 91 has a factor pair, one number in the pair is no bigger than 9, the square root of 91 rounded down. So we only test the primes up to 9.Primes up to 9: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.91 ÷ 2 = 45 remainder 1. Not a factor.
91 ÷ 3 = 30 remainder 1. Not a factor.
91 ÷ 5 = 18 remainder 1. Not a factor.
91 ÷ 7 = 13, with no remainder. So 7 is a factor and 91 is not prime. - Decide. A prime has no factors other than 1 and itself.91 is not prime.
Check. 7 × 13 = 91. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 97 has a factor pair, one number in the pair is no bigger than 9, the square root of 97 rounded down. So we only test the primes up to 9.Primes up to 9: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.97 ÷ 2 = 48 remainder 1. Not a factor.
97 ÷ 3 = 32 remainder 1. Not a factor.
97 ÷ 5 = 19 remainder 2. Not a factor.
97 ÷ 7 = 13 remainder 6. Not a factor. - Decide. A prime has no factors other than 1 and itself.97 is prime.
Check. No prime from 2 to 9 divides into 97, so its only factors are 1 and 97. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Cross out the numbers that have a factor other than 1 and themselves. The ones left are prime.
- Find the numbers that are not prime. Each of these has a factor other than 1 and itself, so cross it out.50 = 2 × 25 51 = 3 × 17 52 = 2 × 26
54 = 2 × 27 55 = 5 × 11 56 = 2 × 28
57 = 3 × 19 58 = 2 × 29 60 = 2 × 30
62 = 2 × 31 63 = 3 × 21 64 = 2 × 32
65 = 5 × 13 66 = 2 × 33 68 = 2 × 34
69 = 3 × 23 70 = 2 × 35 - Check the numbers that are left. Each one has no factor from 2 up to its square root.53: no factor from 2 to 7
59: no factor from 2 to 7
61: no factor from 2 to 7
67: no factor from 2 to 8 - List the numbers that are left. These are the prime numbers.53, 59, 61, 67
Check. There are 4 prime numbers from 50 to 70. ✓
Watch out. 1 is not prime, and it is not crossed out by a factor either. Check it separately.
Watch out
Odd does not mean prime. 9, 15 and 21 are odd, but they are not prime. A number that ends in 5 is not always prime: 55 = 5 × 11. And remember that 1 is not prime.Exercise 3C Prime numbers up to 100
Non-calculatorKey words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 21 has a factor pair, one number in the pair is no bigger than 4, the square root of 21 rounded down. So we only test the primes up to 4.Primes up to 4: 2, 3
- Divide by each prime in turn. Stop as soon as one of them divides exactly.21 ÷ 2 = 10 remainder 1. Not a factor.
21 ÷ 3 = 7, with no remainder. So 3 is a factor and 21 is not prime. - Decide. A prime has no factors other than 1 and itself.21 is not prime.
Check. 3 × 7 = 21. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 23 has a factor pair, one number in the pair is no bigger than 4, the square root of 23 rounded down. So we only test the primes up to 4.Primes up to 4: 2, 3
- Divide by each prime in turn. Stop as soon as one of them divides exactly.23 ÷ 2 = 11 remainder 1. Not a factor.
23 ÷ 3 = 7 remainder 2. Not a factor. - Decide. A prime has no factors other than 1 and itself.23 is prime.
Check. No prime from 2 to 4 divides into 23, so its only factors are 1 and 23. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 27 has a factor pair, one number in the pair is no bigger than 5, the square root of 27 rounded down. So we only test the primes up to 5.Primes up to 5: 2, 3, 5
- Divide by each prime in turn. Stop as soon as one of them divides exactly.27 ÷ 2 = 13 remainder 1. Not a factor.
27 ÷ 3 = 9, with no remainder. So 3 is a factor and 27 is not prime. - Decide. A prime has no factors other than 1 and itself.27 is not prime.
Check. 3 × 9 = 27. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 29 has a factor pair, one number in the pair is no bigger than 5, the square root of 29 rounded down. So we only test the primes up to 5.Primes up to 5: 2, 3, 5
- Divide by each prime in turn. Stop as soon as one of them divides exactly.29 ÷ 2 = 14 remainder 1. Not a factor.
29 ÷ 3 = 9 remainder 2. Not a factor.
29 ÷ 5 = 5 remainder 4. Not a factor. - Decide. A prime has no factors other than 1 and itself.29 is prime.
Check. No prime from 2 to 5 divides into 29, so its only factors are 1 and 29. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 33 has a factor pair, one number in the pair is no bigger than 5, the square root of 33 rounded down. So we only test the primes up to 5.Primes up to 5: 2, 3, 5
- Divide by each prime in turn. Stop as soon as one of them divides exactly.33 ÷ 2 = 16 remainder 1. Not a factor.
33 ÷ 3 = 11, with no remainder. So 3 is a factor and 33 is not prime. - Decide. A prime has no factors other than 1 and itself.33 is not prime.
Check. 3 × 11 = 33. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 37 has a factor pair, one number in the pair is no bigger than 6, the square root of 37 rounded down. So we only test the primes up to 6.Primes up to 6: 2, 3, 5
- Divide by each prime in turn. Stop as soon as one of them divides exactly.37 ÷ 2 = 18 remainder 1. Not a factor.
37 ÷ 3 = 12 remainder 1. Not a factor.
37 ÷ 5 = 7 remainder 2. Not a factor. - Decide. A prime has no factors other than 1 and itself.37 is prime.
Check. No prime from 2 to 6 divides into 37, so its only factors are 1 and 37. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Cross out the numbers that have a factor other than 1 and themselves. The ones left are prime.
- Find the numbers that are not prime. Each of these has a factor other than 1 and itself, so cross it out.30 = 2 × 15 32 = 2 × 16 33 = 3 × 11
34 = 2 × 17 35 = 5 × 7 36 = 2 × 18
38 = 2 × 19 39 = 3 × 13 40 = 2 × 20
42 = 2 × 21 44 = 2 × 22 45 = 3 × 15
46 = 2 × 23 48 = 2 × 24 49 = 7 × 7
50 = 2 × 25 - Check the numbers that are left. Each one has no factor from 2 up to its square root.31: no factor from 2 to 5
37: no factor from 2 to 6
41: no factor from 2 to 6
43: no factor from 2 to 6
47: no factor from 2 to 6 - List the numbers that are left. These are the prime numbers.31, 37, 41, 43, 47
Check. There are 5 prime numbers from 30 to 50. ✓
Watch out. 1 is not prime, and it is not crossed out by a factor either. Check it separately.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 51 has a factor pair, one number in the pair is no bigger than 7, the square root of 51 rounded down. So we only test the primes up to 7.Primes up to 7: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.51 ÷ 2 = 25 remainder 1. Not a factor.
51 ÷ 3 = 17, with no remainder. So 3 is a factor and 51 is not prime. - Decide. A prime has no factors other than 1 and itself.51 is not prime.
Check. 3 × 17 = 51. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 57 has a factor pair, one number in the pair is no bigger than 7, the square root of 57 rounded down. So we only test the primes up to 7.Primes up to 7: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.57 ÷ 2 = 28 remainder 1. Not a factor.
57 ÷ 3 = 19, with no remainder. So 3 is a factor and 57 is not prime. - Decide. A prime has no factors other than 1 and itself.57 is not prime.
Check. 3 × 19 = 57. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 61 has a factor pair, one number in the pair is no bigger than 7, the square root of 61 rounded down. So we only test the primes up to 7.Primes up to 7: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.61 ÷ 2 = 30 remainder 1. Not a factor.
61 ÷ 3 = 20 remainder 1. Not a factor.
61 ÷ 5 = 12 remainder 1. Not a factor.
61 ÷ 7 = 8 remainder 5. Not a factor. - Decide. A prime has no factors other than 1 and itself.61 is prime.
Check. No prime from 2 to 7 divides into 61, so its only factors are 1 and 61. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 63 has a factor pair, one number in the pair is no bigger than 7, the square root of 63 rounded down. So we only test the primes up to 7.Primes up to 7: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.63 ÷ 2 = 31 remainder 1. Not a factor.
63 ÷ 3 = 21, with no remainder. So 3 is a factor and 63 is not prime. - Decide. A prime has no factors other than 1 and itself.63 is not prime.
Check. 3 × 21 = 63. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 71 has a factor pair, one number in the pair is no bigger than 8, the square root of 71 rounded down. So we only test the primes up to 8.Primes up to 8: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.71 ÷ 2 = 35 remainder 1. Not a factor.
71 ÷ 3 = 23 remainder 2. Not a factor.
71 ÷ 5 = 14 remainder 1. Not a factor.
71 ÷ 7 = 10 remainder 1. Not a factor. - Decide. A prime has no factors other than 1 and itself.71 is prime.
Check. No prime from 2 to 8 divides into 71, so its only factors are 1 and 71. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Cross out the numbers that have a factor other than 1 and themselves. The ones left are prime.
- Find the numbers that are not prime. Each of these has a factor other than 1 and itself, so cross it out.54 = 2 × 27 55 = 5 × 11 56 = 2 × 28
57 = 3 × 19 58 = 2 × 29 - Check the numbers that are left. Each one has no factor from 2 up to its square root.59: no factor from 2 to 7
- List the numbers that are left. These are the prime numbers.59
Check. There are 1 prime numbers from 54 to 59. ✓
Watch out. 1 is not prime, and it is not crossed out by a factor either. Check it separately.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Cross out the numbers that have a factor other than 1 and themselves. The ones left are prime.
- Find the numbers that are not prime. Each of these has a factor other than 1 and itself, so cross it out.81 = 3 × 27 82 = 2 × 41
- Check the numbers that are left. Each one has no factor from 2 up to its square root.83: no factor from 2 to 9
- List the numbers that are left. These are the prime numbers.83
Check. There are 1 prime numbers from 81 to 83. ✓
Watch out. 1 is not prime, and it is not crossed out by a factor either. Check it separately.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. List the primes in the range, then count them.
- List the prime numbers from 1 to 20. A prime has exactly two factors: 1 and itself.2, 3, 5, 7, 11, 13, 17, 19
- Count them. Count each prime once.8 primes
Check. 19 is the last prime in the list, and the next number after it, 20, is not prime. ✓
Watch out. Include the end numbers if they are prime. Check the list again before you count.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Think about even numbers.
- Look at the even numbers. Every even number bigger than 2 has 2 as a factor, so it is not prime.4 = 2 × 2, 6 = 2 × 3, 8 = 2 × 4
- Check the number 2. Its only factors are 1 and 2, so it is prime.Factors of 2: 1 and 2
- Fill the gap. 2
Check. 2 is the only even prime, because 2 ÷ 2 = 1 and 2 ÷ 1 = 2. ✓
Watch out. Do not say that all odd numbers are prime. 9 is odd and not prime.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Twin primes are two primes that differ by 2. Add 2 to each prime and check whether the answer is also prime.
- List the primes less than 50. Twin primes must both be on this list.2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
- Look for primes that are 2 apart. Add 2 to each prime. If the answer is also prime, they are twin primes.3 and 5: 5 is prime
5 and 7: 7 is prime
11 and 13: 13 is prime
17 and 19: 19 is prime
29 and 31: 31 is prime
41 and 43: 43 is prime - Write the pairs. Each pair is written once.3 and 5; 5 and 7; 11 and 13; 17 and 19; 29 and 31; 41 and 43
Check. The differences are all 2: 5 − 3 = 2, 7 − 5 = 2, 13 − 11 = 2, and so on. ✓
Watch out. 3 and 5 are twin primes, but 2 and 4 are not, because 4 is not prime.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Find the primes that add up to the total. For each prime, check whether the number left over is also prime.
- Write the primes that are less than the total. Each prime must be less than the total, because the other prime is at least 2.2, 3, 5, 7, 11, 13, 17, 19, 23
- Take each prime away from the total. The other number must also be prime. Check it.30 − 7 = 23, which is prime
30 − 11 = 19, which is prime
30 − 13 = 17, which is prime - Count the pairs. Do not count the same pair twice, such as 7 + 23 and 23 + 7.3 pairs: 7 + 23, 11 + 19, 13 + 17
Check. 7 + 23 = 30 11 + 19 = 30 13 + 17 = 30. ✓
Watch out. Apart from 2, every prime is odd. Two odd numbers add up to an even number, so a pair that includes 2 always has an odd total.
Method. Use the clues one at a time. Each clue crosses out the numbers that do not fit it, until one is left.
- Use the clue: “I am more than 40 and less than 60” Keep only the numbers that fit this clue.41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59
- Use the clue: “I am a prime number” Keep only the numbers that fit this clue.41, 43, 47, 53, 59
- Use the clue: “the sum of my digits is 8” Keep only the numbers that fit this clue.53
- Check the answer against every clue. It must fit all of them.“I am more than 40 and less than 60” ✓
“I am a prime number” ✓
“the sum of my digits is 8” ✓
Check. 53 fits every clue. ✓
Watch out. Read every clue. A number that fits most of them is not the answer.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. A counter-example is one example that breaks the rule. Find an odd number that has a factor other than 1 and itself.
- List the odd numbers greater than 1, in order. Start at 3 and count in odd numbers.3, 5, 7, 9
- Test each one until you find one that is not prime. Look for a factor other than 1 and itself.3: prime, no factor other than 1 and itself
5: prime, no factor other than 1 and itself
7: prime, no factor other than 1 and itself
9: 3 × 3 = 9, so not prime - Write the first counter-example. It is odd, but it is not prime.9 = 3 × 3, so it is not prime
Check. 9 ÷ 3 = 3 with no remainder, so 3 is a factor. ✓
Watch out. Odd does not mean prime. 9 is odd, but it is 3 × 3.
Method. Test some examples. If you find one that breaks the rule, the answer is not “always”.
- Try some examples. Test the statement with numbers you know well.3 + 5 = 8, 7 + 11 = 18, 13 + 17 = 30. Each total is even.
- Explain why. Every prime except 2 is odd. An odd number plus an odd number is always even.Always
Check. The examples agree with the reason. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not.
Method. Test some examples. If you find one that breaks the rule, the answer is not “always”.
- Try some examples. Test the statement with numbers you know well.3 + 5 = 8 is even. But 2 + 3 = 5 is odd.
- Look for an example that breaks the statement. One example that does not fit is enough to show that the statement is not always true.2 + 3 = 5 is odd, so the sum is not always even.
- Explain why. When both primes are odd, the total is even. When one of them is 2, the total is odd.Sometimes
Check. The examples agree with the reason. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Use the definition of a prime: it must have exactly two factors.
- List the factors of 1. The only number that divides into 1 exactly is 1 itself.Factors of 1: 1
- Count the factors. A prime needs exactly two factors, so 1 is not enough.Only one factor
- Decide. So 1 is not a prime number.No
Check. Compare with 2: the factors of 2 are 1 and 2, which is two factors, so 2 is prime. ✓
Watch out. 1 is special: it is neither prime nor composite.