The big picture
Everything in this chapter comes from one fact. Every composite number has one set of primes, apart from order.
| Idea | Remember | Example |
|---|---|---|
| Break into primes | Divide by primes until you reach 1. | 84 = 2 × 2 × 3 × 7 |
| One set of primes | Only the order can change. | 180 = 2 × 2 × 3 × 3 × 5 |
| HCF and LCM | HCF: smallest powers. LCM: greatest powers. | 36 and 60: HCF 12, LCM 180 |
| Squares | A perfect square has an even power of every prime. | 36 = 2² × 3²; 40 is not |
| The lemma | If a prime divides n², it divides n. A divisor that is not prime can fail. | 4 divides 36, but not 6 |
| Proof for √2, √3, √5 | Suppose, square, lemma, substitute, lemma, contradiction. | p² = 5q² leads to 5 dividing q |
| What a proof settles | A search tries some fractions. A proof covers all. | 2.236 × 2.236 misses 5 |
| One example | One example is enough to refuse a claim. | √2 × √2 = 2 |
| A number joined to a root | Suppose the number is a fraction. Get the root alone. The root would be a fraction. | 4 + √3 gives √3 = p/q − 4 |
| Three numbers | Count a missing prime as power 0. HCF takes the smallest power of each prime. LCM takes the greatest. | 18, 30, 45: HCF 3, LCM 90 |
HCF is the highest common factor. LCM is the lowest common multiple. A lemma is a small fact used in bigger proofs. In the lemma row, n is a whole number.
Now open Chapter 1, Real Numbers, in your NCERT book. Start with the exercise on prime factors.