Which Of The Statements Below Are True About Lcm?

Introduction

LCM or Least Common Multiple is a mathematical concept that is used extensively in various fields like physics, chemistry, and computer science. It is a fundamental concept that is taught in schools and colleges. In this article, we will discuss some of the statements related to LCM and find out which of them are true.

Statement 1: LCM is the smallest common multiple of two or more numbers.

This statement is true. LCM is defined as the smallest positive integer that is divisible by all the given numbers. For example, the LCM of 4 and 6 is 12, which is the smallest number that is divisible by both 4 and 6.

Statement 2: LCM can only be calculated for two numbers.

This statement is false. LCM can be calculated for any number of given numbers. For example, the LCM of 2, 3, and 5 is 30.

Statement 3: LCM is always greater than or equal to the largest number in the given set.

This statement is false. LCM can be equal to any of the given numbers. For example, the LCM of 5 and 5 is 5.

Statement 4: LCM is always a multiple of the GCD (Greatest Common Divisor) of the given numbers.

This statement is true. This is known as the fundamental theorem of arithmetic. For example, the LCM of 4 and 6 is 12, and the GCD of 4 and 6 is 2. 12 is a multiple of 2.

Statement 5: LCM of two prime numbers is always their product.

This statement is true. Since prime numbers have no common factors other than 1, their LCM is always their product. For example, the LCM of 2 and 3 is 6.

Statement 6: LCM of two co-prime numbers is always 1.

This statement is false. The LCM of two co-prime numbers is always their product. For example, the LCM of 3 and 5 is 15.

Statement 7: LCM of two consecutive numbers is their product.

This statement is false. The LCM of two consecutive numbers is always the larger number. For example, the LCM of 6 and 7 is 42.

Statement 8: LCM of two numbers is always equal to the product of the two numbers if they are coprime.

This statement is true. If two numbers are co-prime, then their GCD is 1. Therefore, their LCM is simply their product. For example, the LCM of 3 and 5 is 15.

Statement 9: LCM can be used to solve problems related to the time taken by two or more people to complete a task.

This statement is true. LCM can be used to find the time taken by two or more people to complete a task if their individual times are given. For example, if person A takes 4 hours and person B takes 6 hours to complete a task, then the LCM of 4 and 6 is 12. Therefore, they can complete the task together in 12/4 + 12/6 = 3 + 2 = 5 hours.

Statement 10: LCM can be used to calculate the number of times two events occur at the same time.

This statement is true. LCM can be used to find the number of times two events occur at the same time if their individual periods are given. For example, if event A occurs every 4 days and event B occurs every 6 days, then the LCM of 4 and 6 is 12. Therefore, they will occur at the same time every 12 days.

Conclusion

LCM is an important concept in mathematics that has applications in various fields. In this article, we discussed some of the statements related to LCM and found out which of them are true. We hope that this article has helped you understand the concept of LCM better.