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Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Number

Factors, Multiples, Primes, HCF and LCM

Find factors and multiples, test for primes, write numbers as products of prime factors, and use them to find the HCF and LCM in problems.

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

In the curriculum
  • Understand factors, multiples and prime factors; write a number as a product of its prime factors; find the highest common factor (HCF) and the lowest common multiple (LCM) of numbers, and use them to solve problems (8Ni.03)

Every whole number bigger than 1 is built from prime numbers, in the same way that every word is built from letters. On this page you will learn how to break a number into its prime factors, and how to use them to find the highest common factor (HCF) and lowest common multiple (LCM) quickly — even for big numbers. These skills appear on both papers and are used again in fractions, ratio and algebra.

Key words

Word / phraseMeaning / use
factorعامل / قاسم — a whole number that divides exactly into another: 6 is a factor of 24
multipleمضاعف — the result of multiplying by a whole number: 24 is a multiple of 6
prime numberعدد أولي — has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, …
composite numberعدد غير أولي (مؤلف) — has more than two factors: 4, 6, 8, 9, …
prime factorعامل أولي — a factor that is also prime: the prime factors of 12 are 2 and 3
product of prime factorsتحليل العدد إلى عوامله الأولية — 12 = 2 × 2 × 3 = 22 × 3
index formصورة الأسس — writing repeated factors with powers: 2 × 2 × 2 = 23
highest common factor (HCF)القاسم المشترك الأكبر (ق.م.أ) — the biggest number that is a factor of both numbers
lowest common multiple (LCM)المضاعف المشترك الأصغر (م.م.أ) — the smallest number that is a multiple of both numbers

1. Factors and multiples

Find factors in pairs. Start at 1 and work upwards: 1 × 36, 2 × 36… ask "does 2 divide 36? does 3?" Each time you find one factor, you also get its partner. Stop when the numbers meet in the middle. For 36 the pairs are joined like a rainbow:

1234691218361 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
The factors of 36 in pairs. 6 × 6 has no partner, so 36 has an odd number of factors (9). Only square numbers have an odd number of factors.

Multiples go on for ever: the multiples of 7 are 7, 14, 21, 28, 35, … If a is a factor of b, then b is a multiple of a. So "6 is a factor of 42" and "42 is a multiple of 6" say the same thing.

A number is divisible by…if…Example
2the last digit is even (0, 2, 4, 6, 8)538
3the sum of the digits is a multiple of 3471: 4 + 7 + 1 = 12 ✓
4the last two digits make a multiple of 41 316: 16 ✓
5the last digit is 0 or 5905
6it is divisible by 2 and by 3222: even, 2 + 2 + 2 = 6 ✓
8the last three digits make a multiple of 85 120: 120 = 8 × 15 ✓
9the sum of the digits is a multiple of 9729: 7 + 2 + 9 = 18 ✓
10the last digit is 03 470

2. Prime numbers

A prime number has exactly two factors: 1 and itself. The primes less than 50 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

  • 1 is not prime — it has only one factor.
  • 2 is the only even prime. Every other even number has 2 as a factor, so it has more than two factors.
  • To test whether a number is prime, try dividing it by the primes 2, 3, 5, 7, 11, … You only need to try primes up to the square root of the number. For 97: √97 is just under 10, so try 2, 3, 5 and 7. None divides 97, so 97 is prime. For 91: 91 ÷ 7 = 13, so 91 is not prime (91 = 7 × 13).

3. Writing a number as a product of prime factors

Every whole number greater than 1 can be written as a product of primes in exactly one way (apart from the order). There are two methods.

Method 1: factor tree. Split the number into any factor pair, then keep splitting until every branch ends in a prime. Ring the primes as you go.

360103625492233360 = 2 × 2 × 2 × 3 × 3 × 5 = 23 × 32 × 5
A factor tree for 360. Whichever pair you start with (for example 2 × 180 or 6 × 60), you always end with the same primes.

Method 2: repeated division (the ladder). Divide by the smallest prime that goes in exactly, again and again, until you reach 1. Example: 252.

Divide byNumber
2252
2126
363
321
77
1

So 252 = 2 × 2 × 3 × 3 × 7 = 22 × 32 × 7. Always write the primes in order from smallest to largest, and use index form when a question asks for it. Check by multiplying back: 4 × 9 × 7 = 36 × 7 = 252 ✓.

Using prime factors. Once you know 360 = 23 × 32 × 5, you can answer questions without dividing:

  • Is 360 divisible by 12? 12 = 22 × 3, and 22 and 3 are both "inside" 23 × 32 × 5. Yes (360 ÷ 12 = 2 × 3 × 5 = 30).
  • Is 360 divisible by 7? 7 is not one of its prime factors. No.
  • Is 360 divisible by 27? 27 = 33, but 360 only has 32. No.
  • Square numbers have every prime factor to an even power: 144 = 24 × 32. 360 = 23 × 32 × 5 is not square, because 23 and 5 have odd powers.

4. Highest common factor (HCF)

Listing method (small numbers). HCF of 18 and 24: factors of 18 are 1, 2, 3, 6, 9, 18; factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The common factors are 1, 2, 3, 6, so the HCF is 6.

Prime factor method (any numbers). Write both numbers as products of primes. The HCF is the product of the primes they share, using the smaller power of each shared prime.

84 = 22 × 3 × 7 and 90 = 2 × 32 × 5. Shared primes: 2 (smaller power 21) and 3 (smaller power 31). HCF = 2 × 3 = 6.

5. Lowest common multiple (LCM)

Listing method. LCM of 6 and 8: multiples of 6 are 6, 12, 18, 24, 30, …; multiples of 8 are 8, 16, 24, 32, … The first one in both lists is 24.

Prime factor method. The LCM uses every prime that appears in either number, with the larger power of each. For 84 = 22 × 3 × 7 and 90 = 2 × 32 × 5: LCM = 22 × 32 × 5 × 7 = 4 × 9 × 35 = 1260.

6. The Venn diagram method (HCF and LCM together)

Put the prime factors of each number in a circle. Shared primes go in the overlap — once only. Then:

  • HCF = multiply the numbers in the overlap.
  • LCM = multiply every number in the diagram.
Prime factors of 84Prime factors of 90272335HCF = middle only = 2 × 3 = 6LCM = everything = 2 × 7 × 2 × 3 × 3 × 5 = 1260
84 = 2 × 2 × 3 × 7 and 90 = 2 × 3 × 3 × 5. One 2 and one 3 are shared, so they go in the overlap.

A useful generalisation: for two numbers, HCF × LCM = the product of the numbers. Check: 6 × 1260 = 7560 and 84 × 90 = 7560 ✓. This is because every prime factor of the two numbers is used exactly once in HCF × LCM.

ق.م.أ و م.م.أ بسهولة
  • حلّل كل عدد لعوامله الأولية (شجرة العوامل أو القسمة المتكررة).
  • ق.م.أ (HCF): خد العوامل المشتركة بس — بالأس الأصغر. في شكل فن = اللي في النص.
  • م.م.أ (LCM): خد كل العوامل — بالأس الأكبر. في شكل فن = اضرب كل الأرقام اللي في الشكل.
  • الرقم 1 مش عدد أولي، و 2 هو العدد الأولي الزوجي الوحيد.

7. HCF or LCM? Reading the word problem

Use the HCF when the question…Use the LCM when the question…
splits things into the largest equal groups or piecesasks when two repeating events happen together again
asks for the biggest tile, length or group size that fits exactlyasks for the smallest number that both numbers divide into
Answer is smaller than or equal to the numbersAnswer is bigger than or equal to the numbers
Example: share 48 pens and 72 pencils equally into the most bagsExample: two lights flash every 12 s and 18 s
061218243036424854606672minutes after 7:00AB122448601854togethertogether
Bus A leaves every 12 minutes and bus B every 18 minutes. They leave together at 0, 36 and 72 minutes — the common multiples of 12 and 18. The first is the LCM, 36.

Worked examples

Worked example 1 — Write 360 as a product of its prime factors. Give your answer in index form. [2 marks]

  1. Factor tree: 360 = 10 × 36 = (2 × 5) × (4 × 9) = 2 × 5 × 2 × 2 × 3 × 3.
  2. Order the primes: 2 × 2 × 2 × 3 × 3 × 5.
  3. Index form: 23 × 32 × 5.
  4. Check: 8 × 9 × 5 = 72 × 5 = 360 ✓. (Mark scheme style: 1 mark for a correct list of prime factors, e.g. 2, 2, 2, 3, 3, 5; 1 mark for the index form.)

Common mistake: leaving a non-prime in the answer, such as 23 × 9 × 5, or writing it as a sum (2 + 2 + …).

Worked example 2 — Find (a) the HCF and (b) the LCM of 48 and 180. [3 marks]

  1. 48 = 24 × 3 and 180 = 22 × 32 × 5.
  2. (a) Shared primes with the smaller powers: 22 and 3. HCF = 4 × 3 = 12.
  3. (b) Every prime with the larger power: 24, 32 and 5. LCM = 16 × 9 × 5 = 144 × 5 = 720.
  4. Check: HCF × LCM = 12 × 720 = 8640 and 48 × 180 = 8640 ✓.

Common mistake: swapping the rules — using the larger powers for the HCF. The HCF can never be bigger than the smaller number (48).

Worked example 3 — Two buses leave the bus station at 7:00. Bus A leaves every 12 minutes and bus B every 18 minutes. At what time do they next leave together? [2 marks]

  1. "Together again" → LCM of 12 and 18.
  2. Multiples of 12: 12, 24, 36, … Multiples of 18: 18, 36, … LCM = 36.
  3. 7:00 + 36 minutes = 7:36.

Common mistake: giving 36 as the answer — the question asks for a time.

Worked example 4 — Salma has a ribbon 72 cm long and another ribbon 120 cm long. She cuts both ribbons into pieces that are all the same length, with nothing left over. The pieces must be as long as possible. (a) How long is each piece? (b) How many pieces does she get altogether? [3 marks]

  1. "Largest equal pieces" → HCF of 72 and 120.
  2. 72 = 23 × 32 and 120 = 23 × 3 × 5. HCF = 23 × 3 = 24 cm.
  3. (b) 72 ÷ 24 = 3 pieces and 120 ÷ 24 = 5 pieces, so 8 pieces altogether.

Worked example 5 — Is 221 a prime number? Show how you decide. [2 marks]

  1. √221 is between 14 and 15 (142 = 196, 152 = 225), so test the primes 2, 3, 5, 7, 11, 13.
  2. 221 is odd; 2 + 2 + 1 = 5 (not divisible by 3); it does not end in 0 or 5; 221 ÷ 7 = 31.57…; 221 ÷ 11 = 20.09…
  3. 221 ÷ 13 = 17 exactly.
  4. So 221 = 13 × 17 and 221 is not prime.

Common mistake: saying 221 is prime because it is not divisible by 2, 3 or 5. You must test every prime up to the square root.

Worked example 6 — N = 23 × 3 × 52. (a) Is N a multiple of 12? (b) Is N a multiple of 9? (c) Find the smallest whole number you can multiply N by to make a square number. [3 marks]

  1. (a) 12 = 22 × 3. N contains 23 (which includes 22) and 3, so yes.
  2. (b) 9 = 32, but N only contains 31. So no.
  3. (c) A square number has even powers. 23 needs one more 2; 31 needs one more 3; 52 is already even. Multiply by 2 × 3 = 6.
  4. Check: 6N = 24 × 32 × 52 = (22 × 3 × 5)2 = 602 = 3600 ✓ (N = 600 and 600 × 6 = 3600).
Common mistakes
  • Calling 1 a prime number, or forgetting that 2 is prime.
  • Thinking every odd number is prime: 9, 15, 21, 27, 51, 57, 91 are all composite.
  • Leaving composite numbers in a factor tree answer (e.g. 2 × 4 × 9).
  • Mixing up HCF and LCM. The HCF is never bigger than the smaller number; the LCM is never smaller than the bigger number.
  • Thinking the LCM is always the product of the two numbers. That only works when the HCF is 1: LCM(4, 6) = 12, not 24.
  • In a Venn diagram, putting a shared prime in both circles and the overlap — each shared prime goes in the overlap only.
  • Answering a time problem with just the number of minutes.
How Checkpoint asks this
  • "Write 252 as a product of its prime factors" — give a product (×), in index form if asked. Show the tree or ladder: it can earn a part mark.
  • "Find the HCF/LCM of …" — often [1] or [2]. With large numbers, use prime factors or a Venn diagram.
  • Numbers given already in prime-factor form, e.g. "A = 22 × 3 × 7. Find the HCF of A and B" — do not multiply out first; work with the primes.
  • Word problems with buses, lights, tiles, packets or groups — decide HCF or LCM from the table above, and answer in the units the question uses.
  • "Ring the prime numbers" or "Explain why 119 is not prime" — give a factor pair as evidence (119 = 7 × 17).
  • Statements to test: "The LCM of two numbers is always their product." One counter-example proves it false.

Practice

Remember
  • A prime has exactly two factors. 1 is not prime; 2 is the only even prime.
  • Write a number as a product of primes with a factor tree or the ladder, in order, in index form: 360 = 23 × 32 × 5.
  • HCF: shared primes, smaller powers (the overlap of the Venn diagram).
  • LCM: all primes, larger powers (everything in the Venn diagram).
  • Largest equal groups/pieces → HCF. Together again / smallest common number → LCM.
  • Check: HCF × LCM = the product of the two numbers.
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