REAL NUMBERS

Class 10 Maths Formulas

CHAPTER 1 REAL NUMBERS
1.	A number is prime if it has only two factors, 1 and itself.
2.	Every composite number can be expressed as a product of prime factors.
3.	HCF of two numbers = Product of the smaller power of each common factor. HCF of (30, 45) = 3 × 5 = 15 [30 = 2 × 3 × 5 ; 45 = 3² × 5]
4.	LCM of two numbers = Product of the greatest power of each prime factor. LCM of (30, 45) = 2 × 3² × 5 = 90
5.	HCF(a, b) × LCM(a, b) = a × b
6.	HCF(a, b, c) × LCM(a, b, c) ≠ a × b × c

7. Prove that √3 is an irrational number [CBSE 2019 (30/3/1) – Very Important]
Assume √3 is rational. Then √3 = p/q, where q ≠ 0 and p, q are co-primes. √3 = p/q
Squaring both sides: 3 = p² / q² ⇒ p² = 3q²
3 divides p² ⇒ 3 divides p Let p = 3a (for some integer a) ⇒ p² = 9a²
Substitute p² = 3q²: 3q² = 9a² ⇒ q² = 3a² ⇒ 3 divides q
3 divides both p and q. This contradicts the fact that p and q are co-primes.
Therefore, our assumption is wrong. √3 is an irrational number.