Year 6 Maths reasoning (SATs): Factors, multiples and primes
Part of the Year 6 Maths reasoning (SATs) number reasoning chapter. Chapter overview · whole chapter on one page
Factors, multiples and primes
Rule
A factor divides exactly into a number. 36 has the factors 1, 2, 3, 4, 6, 9, 12, 18, 36.
A multiple is in the times table of a number. 12 is a multiple of 3 and of 4.
A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and 13 are primes. 1 is not a prime.
Key ideas
- Factors come in pairs: 4 x 9 = 36, so 4 and 9 are factors of 36.
- The common factors of two numbers are the factors they share.
- A prime number has exactly two factors.
Method. Find pairs of numbers that multiply to 36.
- Start with 1. 1 x 36 = 36, so 1 and 36 are factors.1, 36
- Try 2, 3, 4, 5 and 6. 2 x 18, 3 x 12, 4 x 9 and 6 x 6 work. 5 does not divide 36.2, 18, 3, 12, 4, 9, 6
- Write them in order. Each number is listed once.1, 2, 3, 4, 6, 9, 12, 18, 36
- Write the answer. 1, 2, 3, 4, 6, 9, 12, 18, 36
Check. Each factor divides 36 exactly. ✓
Method. List the factors of each number, then find the largest one in both lists.
- List the factors of 12. 1, 2, 3, 4, 6, 12.1, 2, 3, 4, 6, 12
- List the factors of 18. 1, 2, 3, 6, 9, 18.1, 2, 3, 6, 9, 18
- Find the largest one in both lists. The biggest factor they share is 6.6
- Write the answer. 6
Check. 12 / 6 = 2 and 18 / 6 = 3. ✓
Method. A prime has only two factors, 1 and itself.
- Test 39. 39 = 3 x 13, so it is not prime.39 is not prime
- Test 45 and 51. 45 = 5 x 9 and 51 = 3 x 17, so neither is prime.45 and 51 are not prime
- Test 47. No number from 2 to 6 divides 47 exactly, so its only factors are 1 and 47.47 is prime
- Write the answer. 47
Check. 47 has no factors other than 1 and 47. ✓
Watch out
Watch out. 1 is not a prime number, and 2 is the only even prime.
Exam tip
Practice guide. Check each candidate by dividing by the small primes, 2, 3, 5 and 7, before deciding it is prime.
Exercise 1D Practice: factors, multiples and primes
Non-calculatorMethod. Find pairs that multiply to 24.
- Find pairs. 1 x 24, 2 x 12, 3 x 8, 4 x 6.1, 24, 2, 12, 3, 8, 4, 6
- Write them in order. List each factor once.1, 2, 3, 4, 6, 8, 12, 24
- Write the answer. 1, 2, 3, 4, 6, 8, 12, 24
Check. Each factor divides 24 exactly. ✓
Method. Count in sevens.
- Count in sevens. 7, 14, 21, 28, 35.7 x 1 to 7 x 5
- Write the answer. 7, 14, 21, 28, 35
Check. 7 x 5 = 35. ✓
Method. List the factors of both numbers and find the largest one in both lists.
- Factors of 24. 1, 2, 3, 4, 6, 8, 12, 24.1, 2, 3, 4, 6, 8, 12, 24
- Factors of 36. 1, 2, 3, 4, 6, 9, 12, 18, 36.1, 2, 3, 4, 6, 9, 12, 18, 36
- Find the largest shared factor. The largest number in both lists is 12.12
- Write the answer. 12
Check. 24 / 12 = 2 and 36 / 12 = 3. ✓
Method. List the multiples of each number until you find one in both lists.
- Multiples of 6. 6, 12, 18, 24, 30.6, 12, 18, 24
- Multiples of 8. 8, 16, 24, 32.8, 16, 24
- Find the first shared multiple. 24 is the first number in both lists.24
- Write the answer. 24
Check. 24 / 6 = 4 and 24 / 8 = 3. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. A prime has exactly two factors: 1 and itself.11, 17, 29
- Leave the others unticked. These do not match.9, 15, 21 are not ticked
- Write the answer. 11, 17, 29
Check. Each ticked option is true. ✓
Watch out. 9 = 3 x 3, 15 = 3 x 5 and 21 = 3 x 7, so they are not prime.
Method. Divide 84 by each number and check for a remainder.
- Test 5. 84 / 5 = 16 remainder 4, so 5 is not a factor.no
- Test 6. 84 / 6 = 14, with no remainder.6 is a factor
- Test 8. 84 / 8 = 10 remainder 4, so 8 is not a factor.no
- Write the answer. 6
Check. 6 x 14 = 84. ✓
Method. Divide by the smallest prime that works, and repeat until you reach 1.
- Divide by 2. 60 / 2 = 30.2 x 30
- Divide 30 by 2. 30 / 2 = 15.2 x 2 x 15
- Divide 15 by 3. 15 / 3 = 5, and 5 is prime.2 x 2 x 3 x 5
- Write the answer. 2 x 2 x 3 x 5
Check. 2 x 2 x 3 x 5 = 60. ✓
Method. A prime has no factors except 1 and itself.
- Test 23. 23 is not in the 2, 3 or 5 times tables.23 is prime
- Write the answer. Either 23 or 29 is correct.23
- Write the answer. 23.
Check. 23 is prime. ✓
Method. List the pairs that multiply to 42, then pick the pair with both numbers in the range.
- List the pairs. 1 x 42, 2 x 21, 3 x 14, 6 x 7.6 x 7 = 42
- Check the range. 6 and 7 are both between 5 and 10.6 and 7
- Write the answer. 6 and 7
Check. 6 x 7 = 42. ✓
Method. Find the highest common factor of 48 and 60.
- Factors of 48. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.48 / 12 = 4
- Factors of 60 that are also factors of 48. 12 divides 60 exactly, 5 times.60 / 12 = 5
- Find the largest. The largest shared factor is 12.12
- Write the answer. 12
Check. 5 groups of 12 make 60, and 12 divides both 48 and 60. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. 7 x 13 = 91, so 91 has factors other than 1 and itself.no, because 7 x 13 = 91 fits
- Reject the other choices. The number 91 is odd, so 2 is not a factor, and 2 x 45 is not 91.yes and no, because 2 x 45 = 91 do not fit
- Write the answer. no, because 7 x 13 = 91
Check. no, because 7 x 13 = 91 is the one that fits. ✓
Method. Think about how 6 is made.
- Look at 6. 6 = 2 x 3.6 = 2 x 3
- Check a multiple. 18 = 6 x 3 is also 3 times 6.18 = 3 x 6
- Write the answer. Yes, because 6 is 2 x 3, so any multiple of 6 is 3 times something.
Check. Each multiple of 6 is 6 x a whole number, which is 3 x 2 x that number. ✓
Method. Try two small primes.
- Pick two primes. 2 and 3 are both prime.2, 3
- Add them. 2 + 3 = 5, which is odd.5
- Write the answer. No. 2 + 3 = 5, which is odd.
Check. 5 is odd, so the answer is no. ✓