Chapter 1 · Mathematics
Real Numbers: Chapter Introduction
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Break 360 down and only primes are left, the same ones whichever way you break it. The Fundamental Theorem of Arithmetic, HCF and LCM, and why √2 is irrational.
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Key points
- 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5, whichever factor tree you use.
- Fundamental Theorem of Arithmetic: every composite number is a product of primes, and this factorisation is unique (apart from the order).
- HCF = product of the smallest powers of common primes; LCM = product of the greatest powers of all primes. 12 and 18: HCF 6, LCM 36, and HCF × LCM = 12 × 18.
- √2 cannot be written as p/q (p, q integers, q ≠ 0): it is irrational. The chapter proves √2, √3, √5 irrational by contradiction.
HCF and LCM by prime factorisation
| Number | Prime factors |
|---|---|
| 12 | 2² × 3 |
| 18 | 2 × 3² |
| HCF | 2 × 3 = 6 |
| LCM | 2² × 3² = 36 |
HCF × LCM = 6 × 36 = 216 = 12 × 18
Why √2 is irrational (idea of the proof)
- Suppose √2 = p/q in lowest terms.
- Then p² = 2q², so 2 divides p², hence 2 divides p (a prime dividing a square divides the number).
- Write p = 2r: then q² = 2r², so 2 divides q too.
- p and q both even contradicts lowest terms, so √2 is irrational.
Try it yourself
Goal: Break a number by different routes to test whether the primes ever change, then build the HCF and LCM from them.
- Tap a blue number and choose how to split it, until only primes are left.
- Press Start again and take a different route.
- Try your own number, up to 9999.
- Enter two numbers and compare HCF × LCM with a × b.
Quick quiz
3 questions. Choose an answer for each, then submit to see your score and the explanations.