Every whole number can be broken down into prime building blocks. The Highest Common Factor (HCF) of two numbers is the largest number that divides both of them without leaving any remainder.
To find the HCF of two numbers:
- Express each number as a product of prime numbers.
- Identify the common prime factors.
- For each common prime factor, take the lowest power (smallest exponent).
- Multiply them together!
Example: Find HCF of $72$ and $108$
• $72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$
• $108 = 2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^3$
• Common primes with lowest powers: $2^2 \times 3^2 = 4 \times 9 = \mathbf{36}$
Example: A room has dimensions $18\text{ m}$ by $12\text{ m}$. What is the largest side of a square tile that can pave the room without cutting?
Step 1: The side of the square tile must divide both $18$ and $12$ completely → Find $\text{HCF}(18, 12)$.
Step 2: Factors of $18: 1, 2, 3, 6, 9, 18$; Factors of $12: 1, 2, 3, 4, 6, 12$.
Step 3: $\text{HCF} = \mathbf{6\text{ m}}$.
Number of tiles: $\frac{18 \times 12}{6 \times 6} = 3 \times 2 = \mathbf{6\text{ tiles}}$.
In $2^3 \times 3^2$ and $2^2 \times 3^3$, students see "Highest" in HCF and choose $2^3 \times 3^3$.
Rule: The common factor must fit inside BOTH numbers! You must select the lowest common power ($2^2 \times 3^2 = 36$).
Packaging engineers use HCF to find maximum box sizes that stack evenly inside shipping containers without wasted empty space.