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GCSE Mathematics 01 — Number

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Topics include Types of Numbers, Primes & Factors, Order of Operations & Directed Numbers, Decimals, Rounding & Estimation, Fractions & FDP Conversion, Indices, Powers & Roots, Standard Form, Surds, and Bounds & Error Intervals.

Mathematics EN
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Types of Numbers, Primes & Factors

Understanding the classification of numbers (rational, irrational, integers) and decomposing numbers into prime factors to find HCF and LCM.

Key points

  • Rational numbers can be written as fractions pq\frac{p}{q}; Irrational numbers (like π\pi, 2\sqrt{2}) cannot.
  • Primes have exactly two factors: 1 and themselves. 1 is not prime. 2 is the only even prime.
  • Prime Factorisation: Every integer >1 can be written as a unique product of primes (e.g., using a factor tree).
  • HCF (Highest Common Factor): Product of the lowest power of each common prime factor.
  • LCM (Lowest Common Multiple): Product of the highest power of every prime factor involved.

Worked example

Question

Express 60 and 72 as products of prime factors, then calculate their HCF and LCM.

Solution

1. Prime Factorisation:
60=6×10=(2×3)×(2×5)=22×3×560 = 6 \times 10 = (2 \times 3) \times (2 \times 5) = 2^2 \times 3 \times 5
72=8×9=23×3272 = 8 \times 9 = 2^3 \times 3^2

2. HCF: Take lowest powers of common primes (2 and 3).
22×31=4×3=122^2 \times 3^1 = 4 \times 3 = 12

3. LCM: Take highest powers of all primes (2, 3, 5).
23×32×51=8×9×5=3602^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360

Common pitfalls

  • Thinking the number 1 is a prime number.
  • Confusing factors (divide into the number) with multiples (times table of the number).
  • Calculating HCF instead of LCM for 'next time they flash together' problems.

Prerequisites

  • Multiplication tables
  • Basic division
Further resources