Real Numbers
Real Numbers tests divisibility, prime factorisation, HCF and LCM, irrational numbers, and decimal expansions. RBSE questions often ask for an HCF by Euclid's algorithm, an irrationality proof, or the nature of a decimal.
Core rules
- Euclid's division lemma: for positive integers a and b, a = bq + r with 0 ≤ r < b. Repeated division gives HCF.
- Fundamental Theorem of Arithmetic: every integer greater than 1 is a unique product of primes, apart from order.
- For two positive integers, HCF × LCM = product. For more than two, this relation does not generally hold.
- A rational number p/q in lowest terms has a terminating decimal only when q = 2^m 5^n.
- To prove sqrt(2) irrational, assume sqrt(2)=p/q with p,q coprime; then 2q^2=p^2, so p and q are both even, a contradiction.
Worked example
Find HCF and LCM of 84 and 120. Euclid: 120 = 84×1 + 36; 84 = 36×2 + 12; 36 = 12×3 + 0, so HCF = 12. LCM = (84×120)/12 = 840. Check: 84=2^2×3×7 and 120=2^3×3×5, so HCF=2^2×3=12 and LCM=2^3×3×5×7=840.
Traps
The remainder must satisfy 0 ≤ r < b; a zero remainder ends the algorithm. Do not use HCF × LCM = product for three numbers. In decimal questions, reduce the fraction first. In irrationality proofs, state the coprime assumption and derive the contradiction clearly.