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This page sets out the method for factors & multiples in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Factors & multiples topic page.
When you see 'write down all the factors of …', work from 1 upwards and write each factor pair together.
When two things repeat on different cycles (buses, lights, alarms), list multiples and look for the first match — that is the LCM.
12 = 1 × 12 = 2 × 6 = 3 × 4. Those are all the factor pairs.
Stop when the pairs start to repeat. After 3 × 4 you meet 4 × 3 again.
Every whole number has 1 and itself as factors.
A number whose only factors are 1 and itself is prime. 5 is prime. 1 is not prime, because it has only one factor.
Multiples of 4 are 4, 8, 12, 16… You get them by multiplying 4 by 1, 2, 3, 4…
0 is a multiple of every whole number, but lists of multiples for a question usually start at the number itself.
The factors of 8 are 1, 2, 4, 8. The factors of 12 are 1, 2, 3, 4, 6, 12.
The common factors are 1, 2 and 4. The highest is 4.
Count in the larger number until you hit a multiple of the smaller one.
The lowest common multiple of 4 and 6 is 12.
“Divides exactly”, “goes into” and “factor of” ask for factors.
“In the times table” and “multiple of” ask for multiples.
Hunt factors in pairs so you do not miss the middle. Start at 1, which always works, and write the partner. Then try 2. If the division is exact, write the pair. If it is not, move on. For 18: 1 × 18, 2 × 9, 3 × 6. The next integer, 4, does not divide 18. You can stop once the factors start to repeat, so the full list is 1, 2, 3, 6, 9, 18.
A prime has no pair except 1 and itself. 5 = 1 × 5 only, so 5 is prime. 9 = 3 × 3, so 9 is not prime. 1 has only one factor, and a prime needs two, so 1 is not prime. 2 is the only even prime.
Write factors in order once the pairs are done, and write multiples in order starting at the number itself. The question 'factors of 30' wants 1, 2, 3, 5, 6, 10, 15, 30. The question 'multiples of 30' wants 30, 60, 90 and so on. Swapping those two words is the mistake that turns a short list into a times table, or the other way round.
Method: 1. Start at 1 and write the factor that multiplies with it to make the number. 2. Continue with 2, 3 and so on until the pairs meet. 3. A number is prime when the only pair is 1 and itself.
List every factor of the first number, then every factor of the second, and underline the numbers that appear in both lists. For 12 the factors are 1, 2, 3, 4, 6, 12. For 18 they are 1, 2, 3, 6, 9, 18. The common list is 1, 2, 3, 6. The highest is 6. A smaller common factor such as 2 is common, but it is not the highest, so it does not answer an HCF question.
Check the winner by dividing. 12 ÷ 6 = 2 and 18 ÷ 6 = 3, both exact. Nothing larger works: 12 does not divide 18, and 9 does not divide 12.
Do not confuse the HCF with a common multiple. 36 is a multiple of both 12 and 18, so it is in the LCM conversation. It is larger than both numbers, so it cannot be a factor of either of them. If your 'highest common factor' is bigger than one of the numbers, it is not a factor.
Method: 1. List every factor of the first number and every factor of the second. 2. Underline the factors that appear in both lists. 3. The highest common factor is the largest underlined number.
List multiples of the larger number until you hit a multiple of the smaller one. For 8 and 12, the multiples of 12 are 12, 24, 36, 48. Now test them against 8. 12 ÷ 8 is not exact. 24 ÷ 8 = 3, so 24 is a multiple of both. It is the first one, so the LCM is 24. The multiples of 8 confirm it: 8, 16, 24.
The product of the two numbers is a common multiple, but it is not always the lowest. 8 × 12 = 96, yet 24 is smaller and still common. If the numbers share no factor except 1, the product is the LCM. 4 and 9 give LCM 36.
State the LCM as a number, and show one multiple from each list that lands on it.
Method: 1. Write the multiples of the bigger number first. 2. Stop at the first multiple that is also a multiple of the smaller number. 3. That shared multiple is the lowest common multiple.
Events that repeat and 'next happen together' are an LCM. A school bell rings every 12 minutes. A bus leaves every 18 minutes. They happen together at 8:30. Multiples of 12 and of 18 meet at 36, so they next happen together 36 minutes later, at 9:06. The HCF, 6, is a real shared factor and the wrong tool for this story.
Equal pieces with nothing left over are an HCF. For ropes of 48 cm and 36 cm, the shared factors include 1, 2, 3, 4, 6, 12. The longest piece is 12 cm. That gives 48 ÷ 12 = 4 pieces and 36 ÷ 12 = 3 pieces.
Match the unit to the question. An LCM answer is a time gap, or a clock time if the question asks when. An HCF answer is the piece length. 'How many pieces' means the length divided by that HCF.
Method: 1. A "next time together" question is a lowest common multiple. 2. Sharing into the largest equal groups is a highest common factor. 3. Write the list, then answer in the units the question asked for.
A "product of prime factors" means the number written as primes multiplied together. To find it, divide by the smallest prime that goes in, then keep dividing the answer by primes until you reach 1. For 90: 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. So 90 = 2 × 3 × 3 × 5. A factor tree gives the same result if you keep splitting until every branch ends on a prime.
Index form saves space: 90 = 2 × 3^2 × 5. An exam question might say "write 90 as a product of its prime factors" and give a mark for the method and a mark for the correct answer. Always check by multiplying back. A common slip is to include a number that is not prime, such as 6 or 9, or to write 3 × 3 = 6 by mistake.
Method: 1. Divide by the smallest prime that goes in exactly. 2. Repeat with the answer until you reach 1. 3. List every prime divisor used and write repeats as a power. 4. Multiply back to check you get the original number.
Tip: Start with 2, then 3, then 5, then 7. Most exam numbers only need these four primes.
For large numbers, listing is slow. Write both numbers as products of primes. The HCF takes each prime that appears in both numbers, to the lower power. The LCM takes every prime that appears in either number, to the higher power. For 72 = 2^3 × 3^2 and 108 = 2^2 × 3^3, the HCF is 2^2 × 3^2 = 36 and the LCM is 2^3 × 3^3 = 216.
A Venn diagram shows the same idea. Put the primes the two numbers share in the middle, and the leftover primes in the outer parts. The HCF is the product of the middle. The LCM is the product of everything in the diagram. Exam questions often give you the prime factor form already, for example A = 2^3 × 3 × 5 and B = 2^2 × 3 × 5^2, and ask you to write down the HCF.
Method: 1. Write each number as a product of primes in index form. 2. For the HCF, take the shared primes with the smaller power. 3. For the LCM, take every prime with the larger power. 4. Multiply out and check that each original number divides the LCM exactly.
Quick tests save time. A number is divisible by 2 if it ends in an even digit, by 5 if it ends in 0 or 5, by 3 if its digits add up to a multiple of 3, and by 9 if the digits add up to a multiple of 9. For 51 the digits add to 6, so 51 is a multiple of 3 and it is not prime. 51 = 3 × 17. Many students wrongly call 51 and 91 prime, because they do not test 3 and 7.
Reasoning questions combine clues: "my number is a multiple of 6, a factor of 60, greater than 20 and less than 40". Write the list for one clue, then cross out any numbers that break the others. Multiples of 6 that are factors of 60 are 6, 12, 30 and 60. Only 30 is between 20 and 40. A "show that" or "explain" question needs a calculation as evidence, such as 3 × 17 = 51.
Method: 1. Use the divisibility tests for 2, 3, 5 and 9 first. 2. Write a short list for the first clue. 3. Cross out numbers that break the other clues. 4. Give your evidence as a multiplication or a list.
Repeating events and packs are lowest common multiple questions. Two lights flash every 15 and 20 seconds and start together, so the next time is the LCM, 60 seconds later. A shop sells sausages in packs of 10 and rolls in packs of 6, and you want the same number of each: the LCM is 30, so buy 3 packs of sausages and 5 packs of rolls. The last step, turning the LCM into a number of packs, is the one that loses marks.
Equal groups with nothing left over are highest common factor questions. Tom has 24 pencils and 40 pens and makes identical packs. The most packs possible is the HCF, 8. Each pack then holds 24 ÷ 8 = 3 pencils and 40 ÷ 8 = 5 pens. When a question asks "what time", add the LCM to the start time, taking care with minutes and seconds crossing 60.
Method: 1. Decide whether the story is "next together" (LCM) or "equal groups" (HCF). 2. List multiples or factors, or use prime factors for large numbers. 3. Turn the LCM or HCF into the number the question asked for, such as packs or a clock time. 4. Write the answer with its unit.
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