Solved Examples

Example 1: Which of the following is irrational — √36, √50, 3.14, 22/7? - √36 = 6 → rational (integer) - √50 = 5√2 → irrational (50 is not a perfect square) - 3.14 → terminating decimal → rational - 22/7 → ratio of integers → rational Answer: √50

Example 2: How many prime numbers lie between 50 and 100? Primes in this range: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Answer: 10 primes

Example 3: Is the pair (14, 15) co-prime? Is (14, 21) co-prime? - 14 = 2×7, 15 = 3×5 → no common factor → HCF = 1 → co-prime - 14 = 2×7, 21 = 3×7 → common factor 7 → HCF = 7 → not co-prime

Example 4: Which of the following is composite: 1, 2, 51, 59? - 1 → neither prime nor composite (only one factor) - 2 → prime (only factors 1, 2) - 51 = 3 × 17 → more than two factors → composite - 59 → check divisibility by primes up to √59 ≈ 7.7, i.e., 2, 3, 5, 7 — none divide 59 exactly → prime Answer: 51

Example 5: Are 25, 36, and 49 pairwise co-prime? - 25 = 5², 36 = 2² × 3², 49 = 7² - HCF(25, 36) = 1 (no common prime) → co-prime - HCF(36, 49) = 1 (no common prime) → co-prime - HCF(25, 49) = 1 (no common prime) → co-prime Answer: Yes, all three pairs are co-prime, even though none of the three numbers is itself prime — a good reminder that “co-prime” describes a relationship between numbers, not a property of an individual number.