Number Foundations: HCF and LCM

TL;DR

HCF (Highest Common Factor) is the largest number that divides into two or more numbers without leaving a remainder. LCM (Lowest Common Multiple) is the smallest number that two or more numbers can all divide into. Prime factorisation is a powerful method for finding both HCF and LCM efficiently.

1. The Mental Model

Imagine you have two different-sized pieces of rope. HCF helps you find the longest possible smaller piece you can cut both ropes into without any waste. LCM helps you find the shortest length you'd need to lay out if you wanted to line up both rope lengths perfectly, end-to-end, at the same point.

2. The Core Material

When we talk about factors, we mean numbers that divide exactly into another number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Multiples are what you get when you multiply a number by an integer. For example, the multiples of 3 are 3, 6, 9, 12, 15, and so on.

HCF (Highest Common Factor)

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The HCF is the largest factor that two or more numbers share.

Method 1: Listing Factors
1. List all the factors for each number.
2. Identify the factors that appear in both lists (common factors).
3. The largest of these common factors is the HCF.

Example: Find the HCF of 12 and 18.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
HCF (12, 18) = 6

Method 2: Prime Factorisation
This is often the most efficient method, especially for larger numbers.
1. Find the prime factorisation for each number. (A prime number is a number greater than 1 that has no positive divisors other than 1 and itself, e.g., 2, 3, 5, 7, 11...).
2. Identify the prime factors that are common to all numbers.
3. Multiply these common prime factors, using the lowest power they appear with in any of the factorisations.

Example: Find the HCF of 24 and 36.
Prime factors of 24: $2 \times 2 \times 2 \times 3 = 2^3 \times 3^1$
Prime factors of 36: $2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$
Common prime factors are 2 and 3.
Lowest power of 2 is $2^2$.
Lowest power of 3 is $3^1$.
HCF (24, 36) = $2^2 \times 3^1 = 4 \times 3 = 12$

LCM (Lowest Common Multiple)

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The LCM is the smallest multiple that two or more numbers share.

Method 1: Listing Multiples
1. List the multiples for each number until you find a common one.
2. The first (smallest) common multiple you find is the LCM.

Example: Find the LCM of 4 and 6.
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24, 30...
Common multiples: 12, 24...
LCM (4, 6) = 12

Method 2: Prime Factorisation
This is also very efficient for LCM.
1. Find the prime factorisation for each number.
2. Identify all prime factors that appear in any of the factorisations.
3. Multiply these prime factors, using the highest power they appear with in any of the factorisations.

Example: Find the LCM of 24 and 36.
Prime factors of 24: $2^3 \times 3^1$
Prime factors of 36: $2^2 \times 3^2$
All prime factors are 2 and 3.
Highest power of 2 is $2^3$.
Highest power of 3 is $3^2$.
LCM (24, 36) = $2^3 \times 3^2 = 8 \times 9 = 72$

Relationship between HCF and LCM

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There's a neat relationship for any two numbers:
Product of the two numbers = HCF($a, b$) $\times$ LCM($a, b$)

For example, using 24 and 36:
$24 \times 36 = 864$
HCF (24, 36) = 12
LCM (24, 36) = 72
$12 \times 72 = 864$
It works!

This diagram shows the process for finding HCF and LCM using prime factorisation:

graph TD
    A["Start with Numbers"] --> B["Prime Factorise Each Number"];
    B --> C1["Identify COMMON Prime Factors"];
    B --> C2["Identify ALL Prime Factors"];
    C1 --> D1["For HCF: Take LOWEST power of common factors"];
    C2 --> D2["For LCM: Take HIGHEST power of all factors"];
    D1 --> E1["Multiply lowest powers for HCF"];
    D2 --> E2["Multiply highest powers for LCM"];
    E1 --> F["Result: HCF"];
    E2 --> G["Result: LCM"];

3. Worked Example

Let's find the HCF and LCM of 60 and 72 using prime factorisation.

Step 1: Prime Factorisation
For 60:
$60 = 2 \times 30$
$30 = 2 \times 15$
$15 = 3 \times 5$
So, $60 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3^1 \times 5^1$

For 72:
$72 = 2 \times 36$
$36 = 2 \times 18$
$18 = 2 \times 9$
$9 = 3 \times 3$
So, $72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$

Step 2: Find HCF
Common prime factors are 2 and 3.
Lowest power of 2: $2^2$ (from 60)
Lowest power of 3: $3^1$ (from 60)
HCF (60, 72) = $2^2 \times 3^1 = 4 \times 3 = 12$

Step 3: Find LCM
All prime factors involved are 2, 3, and 5.
Highest power of 2: $2^3$ (from 72)
Highest power of 3: $3^2$ (from 72)
Highest power of 5: $5^1$ (from 60)
LCM (60, 72) = $2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 72 \times 5 = 360$

Check:
$60 \times 72 = 4320$
$HCF \times LCM = 12 \times 360 = 4320$
The results are consistent.

4. Key Takeaways

• HCF is the largest number that divides evenly into all given numbers.
• LCM is the smallest number that all given numbers can divide evenly into.
• Prime factorisation is generally the most reliable method for finding both HCF and LCM.
• For HCF using prime factors, you take the lowest power of the common prime factors.
• For LCM using prime factors, you take the highest power of all prime factors present.
• The product of two numbers equals the product of their HCF and LCM.

Common Mistakes to Avoid:
- Mixing up 'lowest power' for HCF and 'highest power' for LCM during prime factorisation.
- Forgetting to include all unique prime factors when calculating LCM using prime factorisation.
- Incorrectly finding prime factors for the initial numbers.
- Trying to list all factors or multiples for very large numbers, which can be time-consuming and prone to error.

5. Now Try It

Find the HCF and LCM of 48, 72, and 108. Use the prime factorisation method.

What to do:
1. Find the prime factorisation for 48, 72, and 108.
2. Use these factorisations to calculate the HCF of all three numbers.
3. Use the same factorisations to calculate the LCM of all three numbers.

What success looks like:
You should arrive at HCF = 12 and LCM = 432.