Find the HCF and the LCM of 18 and 30.
- 18 = 2 × 32 and 30 = 2 × 3 × 5.
- HCF: common primes with the lowest powers = 2 × 3 = 6.
- LCM: all primes with the highest powers = 2 × 32 × 5 = 90.
Answer: HCF = 6, LCM = 90
Types of number
Maths · Number. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
Numbers are sorted into families, and questions use the family names. Integers are the whole numbers, positive, negative and zero. A rational number can be written as a fraction of two integers, so every terminating or recurring decimal is rational. An irrational number, such as π or √2, cannot be written as a fraction.
A prime number has exactly two factors, 1 and itself. Every whole number greater than 1 is either prime or a product of primes, and writing that product is the best way to find a highest common factor (HCF) or lowest common multiple (LCM).
Square numbers, cube numbers and reciprocals also belong to this topic.
Rules to remember
- Natural numbers are the counting numbers 1, 2, 3, …; integers are …, −2, −1, 0, 1, 2, …
- A prime has exactly two factors. 1 is not prime and 2 is the only even prime.
- To test whether a number is prime, try dividing by the primes up to its square root.
- HCF = product of the common prime factors, each with its lowest power.
- LCM = product of every prime factor that appears, each with its highest power.
- In a square number every prime factor has an even power.
- reciprocal of n = 1 ÷ n, so the reciprocal of a/b is b/a
Common mistake
Students mix up HCF and LCM. The HCF is never bigger than the smaller number; the LCM is never smaller than the larger number. Use the LCM for questions about events that happen together again.
Worked example
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