Numerical: LCM and HCF MCQ — 15 Practice Questions with Answers

Explanation

For any two positive integers, HCF × LCM = product of the two numbers. This is the standard relation used when one number is missing. Here, HCF = 12 and LCM = 240, so product of the two numbers = 12 × 240 = 2880. One number is 48. Therefore, the other number = 2880 ÷ 48. Since 48 × 60 = 2880, the other number is 60. Hence the correct statement is that the other number is 60.

Explanation

Use prime factorisation to find the HCF. Here, 84 = 2² × 3 × 7, 126 = 2 × 3² × 7 and 210 = 2 × 3 × 5 × 7. For the HCF, take only the common prime factors with the smallest powers. The common factors are 2, 3 and 7. Therefore, HCF = 2 × 3 × 7 = 42. Hence, the highest common factor of 84, 126 and 210 is 42.

Explanation

Prime factorise the two numbers: 18 = 2 × 3² and 24 = 2³ × 3. The HCF uses the smaller powers, so HCF = 2 × 3 = 6. The LCM uses the larger powers, so LCM = 2³ × 3² = 8 × 9 = 72. Now HCF × LCM = 6 × 72 = 432, which is also equal to 18 × 24. Therefore, all three statements are correct.

Explanation

The bells will ring together again after the LCM of their time intervals. Factorise the intervals: 12 = 2² × 3, 15 = 3 × 5 and 20 = 2² × 5. The LCM takes the highest powers of all prime factors, so LCM = 2² × 3 × 5 = 60 minutes. Starting from 8:00, adding 60 minutes gives 9:00. Therefore, the statement saying 8:45 is incorrect.

Explanation

If a number leaves the same remainder 5 on division by 12, 18 and 30, then subtracting 5 from it must give a common multiple of 12, 18 and 30. First find the LCM: 12 = 2² × 3, 18 = 2 × 3², and 30 = 2 × 3 × 5. So LCM = 2² × 3² × 5 = 180. The least required number is 180 + 5 = 185. Therefore the statement saying that the least such number is 175 is incorrect.