What is the difference between HCF and LCM in Mathematics?

HCF, or Highest Common Factor, is the largest number that divides two or more numbers without leaving a remainder. LCM, or Least Common Multiple, is the smallest number that is a multiple of two or more numbers. Both concepts are fundamental in the study of number theory within the IB MYP Mathematics curriculum.

How do you find the HCF of two numbers using prime factorization?

To find the HCF using prime factorization, first express each number as a product of prime factors. Then, identify the common prime factors and multiply them together. The product is the HCF. This method is particularly useful for understanding the structure of numbers in the IB MYP Mathematics program.

Can you explain how to calculate the LCM of two numbers using the listing method?

To calculate the LCM using the listing method, list the multiples of each number until you find the smallest common multiple. This approach helps students visualize the concept of multiples, which is a key learning objective in the IB MYP Mathematics curriculum.

Why is understanding HCF and LCM important in solving real-world problems?

Understanding HCF and LCM is crucial for solving problems involving ratios, proportions, and scheduling. For example, determining the optimal intervals for events or resources often requires calculating LCM, while simplifying fractions involves finding the HCF. These skills are essential in the IB MYP Mathematics framework for applying mathematical concepts to real-life situations.

What are some common mistakes students make when calculating HCF and LCM?

Common mistakes include confusing HCF with LCM, not correctly identifying all prime factors, and miscalculating multiples. To avoid these errors, students should practice different methods and verify their results. Mastery of these concepts is important for success in the IB MYP Mathematics curriculum.

Why is "Confusing HCF with LCM in word problems." a common mistake in HCF and LCM?

Why it happens: Both involve 'common'. How to avoid it: HCF — largest number that BOTH divide INTO (so smaller than the numbers). LCM — smallest number BOTH divide INTO (so bigger than the numbers).

Why is "Taking maximum powers for HCF." a common mistake in HCF and LCM?

Why it happens: Mixing up the rules. How to avoid it: HCF — MIN powers (shared by all). LCM — MAX powers (so every number divides in).

Why is "Listing factors/multiples and missing some." a common mistake in HCF and LCM?

Why it happens: Slow, error-prone method. How to avoid it: Use PRIME FACTORISATION for any number 20. Much faster and reliable.

NumberIB MYP Maths Extended Level

HCF and LCM

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HCF and LCM — IB MYP Mathematics (Extended): highest common factor and lowest common multiple

HCF and LCM unlock fractions, ratios and word problems. This MYP Mathematics Extended note covers prime factorisation, the venn-diagram method, and applications.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Find HCF and LCM using prime factorisation.
MYP Mathematics A — Apply the relationship $\text{HCF} \times \text{LCM} = ab$ .
MYP Mathematics C — Use Venn-diagram or list methods clearly.
MYP Mathematics D — Apply HCF / LCM to scheduling, packaging, ratios.

HCF and LCM: definitions

Factors go down; multiples go up.

FACTORS of a number are the numbers that divide into it. Factors of 12: $1, 2, 3, 4, 6, 12$ .

MULTIPLES of a number are what you get by multiplying it by integers. Multiples of 4: $4, 8, 12, 16, 20, \ldots$

The HCF (Highest Common Factor) of two numbers is the largest number that appears in BOTH their factor lists.

The LCM (Lowest Common Multiple) of two numbers is the smallest number that appears in BOTH their multiple lists.

Numbers	HCF	LCM
12, 18	6	36
8, 12	4	24
15, 25	5	75
7, 11	1	77

(When HCF = 1, the numbers are called coprime. Two coprime numbers have LCM = their product.)

Factor: divides INTO the number.
Multiple: comes OUT of the number when multiplied.
HCF = largest shared factor. LCM = smallest shared multiple.

Prime factorisation method

Break each number into primes, then combine.

Every integer > 1 can be written as a unique product of primes (Fundamental Theorem of Arithmetic).

Step 1 — find prime factorisations.

$72 = 2^3 \times 3^2$ $60 = 2^2 \times 3 \times 5$

Step 2 — for HCF, take the LOWEST power of each prime that appears in BOTH:

Both have $2$ : lowest power is $2^2$ .
Both have $3$ : lowest power is $3^1$ .
Only 60 has 5: don't include.

$\text{HCF}(72, 60) = 2^2 \times 3 = 12.$

Step 3 — for LCM, take the HIGHEST power of each prime that appears in EITHER:

$2$ : highest is $2^3$ .
$3$ : highest is $3^2$ .
$5$ : highest is $5^1$ .

$\text{LCM}(72, 60) = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360.$

Check: $\text{HCF} \times \text{LCM} = 12 \times 360 = 4320$ , and $72 \times 60 = 4320$ . Consistent.

Venn-diagram method: write prime factors in the overlap if shared, in left/right only if not. HCF = product of intersection. LCM = product of everything in the diagram.

The Venn method works for any number of numbers — just use more circles.

HCF: take MINIMUM power of each shared prime.
LCM: take MAXIMUM power of each prime in either.
Venn diagram: overlap = HCF, union = LCM.
Check: $\text{HCF} \times \text{LCM} = a \times b$ .

When to use HCF and when to use LCM

Sharing into equal groups vs synchronising events.

Use HCF when:

Dividing items into the LARGEST possible equal groups.
Simplifying fractions.
Finding the largest square that fits a rectangle.

Use LCM when:

Synchronising recurring events (when will they next coincide?).
Adding/subtracting fractions (common denominator).
Packaging that must tile both an item and a container.

Worked example (HCF). A teacher has 24 pencils and 18 rulers. She wants to make IDENTICAL kits with no items left over. What is the largest number of kits she can make?

The number must divide BOTH 24 and 18.
$\text{HCF}(24, 18) = 6$ .
So she can make 6 identical kits (each with 4 pencils and 3 rulers).

Worked example (LCM). Lights at a junction blink at 6-second and 10-second intervals. They blink together at time 0. When do they next blink together?

The next coincidence is the LCM of the cycle times.
$\text{LCM}(6, 10) = 30$ .
So they blink together every 30 seconds.

This is a classic MYP criterion-D question — picking the right tool (HCF or LCM) for the context is half the work.

HCF: 'biggest equal groups', 'simplify fraction'.
LCM: 'next time both happen', 'common denominator'.
Read the question carefully — pick the right tool.

How it’s examined

Criterion A: HCF/LCM by prime factorisation or Venn. Criterion C: clear diagram and working. Criterion D: pick the right tool for a real-context word problem.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — HCF and LCM

Step-by-step solutions to past-paper-style questions on hcf and lcm, written exactly the way a tutor would explain them at the board.

1HCF by prime factorisation
Building confidence• HCF
Question
Find $\text{HCF}(48, 60)$ .
Step-by-step solution
1. Step 1
  Prime factorise: $48 = 2^4 \times 3$ and $60 = 2^2 \times 3 \times 5$ .
2. Step 2
  Shared primes: $2$ and $3$ .
3. Step 3
  Take minimum powers: $2^2$ (from 60) and $3^1$ (both have $3$ ).
4. Step 4
  $\text{HCF} = 2^2 \times 3 = 12$ .
Answer
$12$
2LCM by prime factorisation
Building confidence• LCM
Question
Find $\text{LCM}(48, 60)$ .
Step-by-step solution
1. Step 1
  Prime factorise: $48 = 2^4 \times 3$ and $60 = 2^2 \times 3 \times 5$ .
2. Step 2
  Take MAXIMUM power of every prime that appears: $2^4$ , $3$ , $5$ .
3. Step 3
  $\text{LCM} = 2^4 \times 3 \times 5 = 16 \times 15 = 240$ .
4. Step 4
  Check: $\text{HCF} \times \text{LCM} = 12 \times 240 = 2880 = 48 \times 60$ . ✓
Answer
$240$
3Use HCF for grouping
Building confidence• application, HCF
Question
A bakery has 36 cupcakes and 24 cookies. They want to make identical gift boxes with all items used. What is the largest number of boxes possible?
Step-by-step solution
1. Step 1
  Number of boxes must divide BOTH 36 and 24 — looking for the LARGEST common divisor: HCF.
2. Step 2
  Prime factorise: $36 = 2^2 \times 3^2$ , $24 = 2^3 \times 3$ .
3. Step 3
  HCF: take min powers: $2^2 \times 3 = 12$ .
4. Step 4
  12 boxes — each has $36/12 = 3$ cupcakes and $24/12 = 2$ cookies.
Answer
12 boxes (each with 3 cupcakes + 2 cookies).
4Use LCM to synchronise
Stretch• application, LCM
Question
Two buses leave the station together. Bus A returns every 36 minutes; Bus B every 48 minutes. When will they next leave together?
Step-by-step solution
1. Step 1
  Next coincidence is the LCM of the cycle times.
2. Step 2
  Prime factorise: $36 = 2^2 \times 3^2$ , $48 = 2^4 \times 3$ .
3. Step 3
  LCM: max powers: $2^4 \times 3^2 = 16 \times 9 = 144$ minutes.
4. Step 4
  $144$ minutes $= 2$ hours $24$ minutes.
Answer
After 144 minutes (2 h 24 min).

Key Definitions and Keywords — HCF and LCM

Definitions to memorise and the exact keywords mark schemes credit for hcf and lcm answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Factor
Examiner keyword
An integer that divides another integer exactly (no remainder).
Multiple
Examiner keyword
A number obtained by multiplying an integer by another integer.
HCF (highest common factor)
Examiner keyword
The largest integer that is a factor of two or more numbers.
LCM (lowest common multiple)
Examiner keyword
The smallest positive integer that is a multiple of two or more numbers.
Prime number
An integer greater than 1 with exactly two factors: 1 and itself.
Coprime
Two integers with HCF $= 1$ . Their LCM equals their product.

Common Mistakes and Misconceptions — HCF and LCM

The traps other students keep falling into on hcf and lcm questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Confusing HCF with LCM in word problems.
Why it happens
Both involve 'common'.
How to avoid it
HCF — largest number that BOTH divide INTO (so smaller than the numbers). LCM — smallest number BOTH divide INTO (so bigger than the numbers).
✕Taking maximum powers for HCF.
Why it happens
Mixing up the rules.
How to avoid it
HCF — MIN powers (shared by all). LCM — MAX powers (so every number divides in).
✕Listing factors/multiples and missing some.
Why it happens
Slow, error-prone method.
How to avoid it
Use PRIME FACTORISATION for any number $\ge 20$ . Much faster and reliable.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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HCF and LCM — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Numbers

HCF

LCM

12, 18

8, 12

15, 25

7, 11

Every integer > 1 can be written as a unique product of primes (Fundamental Theorem of Arithmetic).

Step 1 — find prime factorisations.

$72 = 2^3 \times 3^2$ $60 = 2^2 \times 3 \times 5$

Step 2 — for HCF, take the LOWEST power of each prime that appears in BOTH:

Both have $2$ : lowest power is $2^2$ .
Both have $3$ : lowest power is $3^1$ .
Only 60 has 5: don't include.

$\text{HCF}(72, 60) = 2^2 \times 3 = 12.$

Step 3 — for LCM, take the HIGHEST power of each prime that appears in EITHER:

$2$ : highest is $2^3$ .
$3$ : highest is $3^2$ .
$5$ : highest is $5^1$ .

$\text{LCM}(72, 60) = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360.$

Check: $\text{HCF} \times \text{LCM} = 12 \times 360 = 4320$ , and $72 \times 60 = 4320$ . Consistent.

Venn-diagram method: write prime factors in the overlap if shared, in left/right only if not. HCF = product of intersection. LCM = product of everything in the diagram.

The Venn method works for any number of numbers — just use more circles.

HCF and LCM

Short Study Notes in the form of Slides

Short Study Notes — HCF and LCM

Detailed Notes

What you’ll learn

How it’s examined

1HCF by prime factorisation

2LCM by prime factorisation

3Use HCF for grouping

4Use LCM to synchronise

Factor

Multiple

HCF (highest common factor)

LCM (lowest common multiple)

Prime number

Coprime

✕Confusing HCF with LCM in word problems.

✕Taking maximum powers for HCF.

✕Listing factors/multiples and missing some.

Practice questions

Past paper style quiz

Assess your understanding

HCF and LCM — frequently asked questions

HCF and LCM

Short Study Notes in the form of Slides

Short Study Notes — HCF and LCM

Detailed Notes

What you’ll learn

How it’s examined

1HCF by prime factorisation

2LCM by prime factorisation

3Use HCF for grouping

4Use LCM to synchronise

Factor

Multiple

HCF (highest common factor)

LCM (lowest common multiple)

Prime number

Coprime

✕Confusing HCF with LCM in word problems.

✕Taking maximum powers for HCF.

✕Listing factors/multiples and missing some.

Practice questions

Past paper style quiz

Assess your understanding

HCF and LCM — frequently asked questions