Types of number
Types of number
- natural numbers = counting numbers $1, 2, 3, \dots$; integers = whole numbers (negative, zero, positive).
- a factor divides exactly; a multiple is the number × an integer.
- a prime has exactly two factors (1 and itself); 1 is not prime.
- rational = can be written $\tfrac{a}{b}$; irrational (e.g. $\pi$, $\sqrt2$) cannot.
Practice
Which of these is a prime number?
13 has exactly two factors (1 and 13). 1 is not prime; 9 = 3×3; 15 = 3×5.
Prime factors, HCF and LCM
- Write a number as a product of primes: $72 = 2^3 \times 3^2$, $120 = 2^3 \times 3 \times 5$.
- HCF: lowest power of each shared prime → $2^3 \times 3 = 24$.
- LCM: highest power of every prime → $2^3 \times 3^2 \times 5 = 360$.
Practice
Find the HCF of 72 and 120. (72 = 2³×3², 120 = 2³×3×5)
Take the lowest shared powers: 2³ × 3 = 8 × 3 = 24.
Practice
Find the LCM of 72 and 120.
Take the highest powers: 2³ × 3² × 5 = 8 × 9 × 5 = 360.
You've got it
Key idea
- prime = exactly two factors (1 isn't prime); rational $=\tfrac{a}{b}$, irrational can't be
- HCF = lowest shared prime powers; LCM = highest prime powers
- $72=2^3\times3^2$, $120=2^3\times3\times5$ → HCF $24$, LCM $360$