Primes, factors, multiples — HCF and LCM
OCR J560 Paper 1 (non-calculator) regularly sets HCF/LCM questions. The prime factor tree method is reliable and examiners like to see it. Questions often link HCF/LCM to real-life contexts ("cogs mesh", "buses arrive together").
Prime numbers
A prime number has exactly two distinct factors: 1 and itself. The first ten: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Important: 1 is NOT prime (only one factor). 2 is the only even prime.
Prime factor decomposition
Every integer > 1 can be written as a product of primes (Fundamental Theorem of Arithmetic).
Method: build a factor tree; keep splitting until all branches end in primes.
Example: 360
360 = 2 × 180 = 2 × 2 × 90 = 2 × 2 × 2 × 45 = 2 × 2 × 2 × 3 × 15 = 2 × 2 × 2 × 3 × 3 × 5
So 360 = 2³ × 3² × 5.
Highest Common Factor (HCF)
The HCF of two numbers is the largest number that divides exactly into both.
Method (using prime factors): multiply together all primes that appear in BOTH factorizations, each to the lower power.
Example: HCF(360, 504)?
- 360 = 2³ × 3² × 5
- 504 = 2³ × 3² × 7
- Common: 2³ × 3² = 8 × 9 = 72
Lowest Common Multiple (LCM)
The LCM of two numbers is the smallest number that both divide exactly into.
Method: multiply together all primes that appear in either factorization, each to the higher power.
Example: LCM(360, 504)?
- 360 = 2³ × 3² × 5
- 504 = 2³ × 3² × 7
- All primes at highest power: 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2520
Memory trick: HCF = use LOWER powers, take intersection. LCM = use HIGHER powers, take union.
Real-life applications
- HCF: "What is the largest square tile that fits exactly along a 360 cm and 504 cm wall?" → HCF = 72 cm.
- LCM: "Bus A every 12 mins, Bus B every 20 mins — when do they next both arrive together?" → LCM(12, 20) = 60 mins.
Common OCR exam mistakes
- Confusing HCF and LCM — "highest" is HCF but HCF is the SMALLER number; LCM is LARGER.
- Stopping the factor tree before reaching primes — check every branch ends in a prime.
- Forgetting to use prime factor method for large numbers — trial division becomes very slow.
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