🔢 Numbers Unlocked! Master Integers, Primes & More

Numbers are the building blocks of mathematics. From **counting apples** to **calculating satellite trajectories**, understanding different _types_ of numbers—and how to work with their Highest Common Factor (HCF) and Lowest Common Multiple (LCM)—is crucial for success at O Level and beyond.

Main Concept: The Number Family

Below are the everyday stars of the number universe. Notice how each set has its own special properties, yet they often *overlap*. • **Integers**

(\mathbb{Z})

\{\dots,-3,-2,-1,0,1,2,3,\dots\}

• **Prime numbers**: integers greater than 1 whose only factors are 1 and itself (e.g.

2,3,5,7,11,13,17\dots

) • **Square numbers**: results of

n^2

(e.g.

1,4,9,16

) while **cube numbers** come from

n^3

(e.g.

1,8,27

). • **Rational numbers** can be expressed as a fraction

\frac{p}{q}

where

p,\,q\in\mathbb{Z},\,q\neq 0

. **Irrational numbers** _cannot_, e.g.

\pi

\sqrt{2}

. HCF gives the **largest** shared factor, while LCM provides the **smallest** shared multiple of two or more numbers.

HCF is useful for simplifying fractions.
LCM helps when finding common denominators.
Prime factorisation is the fastest route to both HCF and LCM.

Worked Example: HCF & LCM of 24 and 90

Problem: Find the HCF and LCM of 24 and 90 using prime factorisation.

Step 1: Prime-factorise.
$24 = 2^3 \times 3^1, \quad 90 = 2^1 \times 3^2 \times 5^1$
Step 2: HCF ‑ Take the **lowest** power of each common prime.
$\text{HCF} = 2^{\min(3,1)} \times 3^{\min(1,2)} = 2^1 \times 3^1 = 6$
Step 3: LCM ‑ Take the **highest** power of every prime present.
$\text{LCM} = 2^{3} \times 3^{2} \times 5^{1} = 8 \times 9 \times 5 = 360$
Answer: $\text{HCF}=6$ and $\text{LCM}=360$ .

By firmly grasping these number types and the HCF/LCM toolkit, you'll simplify algebra, conquer fractions, and breeze through non-calculator papers. **Keep practising**, and challenge yourself with bigger numbers to build speed and confidence!