Types of Number, Factors, Primes, HCF and LCM

Types of number

Type	Meaning	Examples
natural numbers	counting numbers	1, 2, 3, ...
integers	whole numbers, including negatives	−3, 0, 7
rational numbers	can be written as a fraction p/q	3/4, −2, 0.25, 0.333...
irrational numbers	cannot be written as a fraction	√2, π
prime numbers	exactly two factors, 1 and itself	2, 3, 5, 7, 11
square numbers	n × n	1, 4, 9, 16
cube numbers	n × n × n	1, 8, 27, 64

⚠️ 1 is not prime. 2 is the only even prime.

Factors and multiples

• A factor of 12 divides 12 exactly: 1, 2, 3, 4, 6, 12.
• A multiple of 12 is in its times table: 12, 24, 36, ...

Product of prime factors

Use a factor tree or repeated division.

Example. 360 = 2 × 180 = 2 × 2 × 90 = ... = 2³ × 3² × 5

HCF and LCM

Write both numbers as products of primes.

• HCF (highest common factor): multiply the common primes using the lowest powers.
• LCM (lowest common multiple): multiply all primes using the highest powers.

Example. 360 = 2³ × 3² × 5 and 84 = 2² × 3 × 7.

HCF = 2² × 3 = 12. LCM = 2³ × 3² × 5 × 7 = 2520.

Word problems

• When will two buses next leave together? : LCM.
• What is the largest equal group size? : HCF.

Quick summary

• Know the number types; 1 is not prime.
• Prime factorisation is the key tool.
• HCF: common primes, lowest powers. LCM: all primes, highest powers.