Real Numbers
What this domain tests
Real Numbers checks whether you can move between Euclid's division lemma, prime factorisation, HCF and LCM, and the proof that certain numbers are irrational. AP SSC questions are usually short: find an HCF, classify a decimal expansion, or complete a contradiction proof.
Core rules
- Euclid's division lemma: for positive integers a and b there are unique integers q and r with a = bq + r and 0 ≤ r < b. Repeating it gives the HCF.
- Fundamental Theorem of Arithmetic: every integer greater than 1 is a product of primes, unique apart from order.
- For two positive integers, HCF(a,b) × LCM(a,b) = a × b. This is a two-number rule; do not apply it to three numbers.
- A rational number p/q in lowest terms terminates exactly when q = 2^m 5^n. Otherwise the decimal recurs.
- √2, √3, √5 and similar numbers are irrational; the standard proof assumes p/q in lowest terms and forces a divisibility contradiction.
Worked example
Find HCF(84, 120). 120 = 84 × 1 + 36 84 = 36 × 2 + 12 36 = 12 × 3 + 0 The last non-zero remainder is 12, so HCF = 12. Then LCM = (84 × 120)/12 = 840.
Traps
- Allowing r = b; the remainder must be strictly less than the divisor.
- Calling a decimal irrational because it is long. The test is termination or recurrence, not length.
- Using the HCF–LCM product rule on three numbers.
- In irrationality proofs, forgetting that p and q are coprime, which is needed for the contradiction.