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Introduction
LCM, or Least Common Multiple, is a mathematical concept that is used in various fields, such as engineering, physics, and computer science. It is the smallest multiple that is divisible by two or more numbers without leaving any remainder. In this article, we will explore some statements about LCM and determine which ones are true.
Statement 1: LCM is only applicable to integers
True. LCM is only applicable to integers, which are whole numbers that do not have any fractional or decimal parts. This is because the concept of divisibility only applies to integers.
Statement 2: LCM is always greater than or equal to the numbers being multiplied
True. LCM is always greater than or equal to the numbers being multiplied. This is because the LCM is a multiple of the numbers being multiplied, and any multiple of a number is greater than or equal to the number itself.
Statement 3: LCM is the same as the product of the numbers being multiplied
False. LCM is not always the same as the product of the numbers being multiplied. It is only the same when the numbers being multiplied are relatively prime, which means they have no common factors other than 1.
Statement 4: LCM can be used to find the highest common factor (HCF) of two or more numbers
True. LCM can be used to find the HCF of two or more numbers using the formula HCF x LCM = Product of the Numbers. This is known as the fundamental theorem of arithmetic.
Statement 5: LCM is commutative
True. LCM is commutative, which means the order of the numbers being multiplied does not affect the LCM. For example, LCM(2,3) = LCM(3,2) = 6.
Statement 6: LCM is associative
True. LCM is associative, which means the grouping of the numbers being multiplied does not affect the LCM. For example, LCM(2,3,4) = LCM(2,4,3) = LCM(3,2,4) = LCM(3,4,2) = LCM(4,2,3) = LCM(4,3,2) = 12.
Statement 7: LCM of two prime numbers is their product
True. LCM of two prime numbers is their product because prime numbers have no common factors other than 1, and their product is the smallest multiple that is divisible by both of them.
Statement 8: LCM of two consecutive numbers is their product
False. LCM of two consecutive numbers is always equal to the larger number because the larger number is already a multiple of the smaller number.
Statement 9: LCM of two co-prime numbers is their product
True. LCM of two co-prime numbers is their product because co-prime numbers have no common factors other than 1, and their product is the smallest multiple that is divisible by both of them.
Statement 10: LCM of two numbers is always greater than or equal to their HCF
True. LCM of two numbers is always greater than or equal to their HCF because the HCF is a factor of the LCM. In other words, the LCM is a multiple of the HCF.
Statement 11: LCM of three numbers is always greater than or equal to their HCF
True. LCM of three numbers is always greater than or equal to their HCF because the HCF is a factor of the LCM. In other words, the LCM is a multiple of the HCF.
Statement 12: LCM of four or more numbers is always greater than or equal to their HCF
True. LCM of four or more numbers is always greater than or equal to their HCF because the HCF is a factor of the LCM. In other words, the LCM is a multiple of the HCF.
Statement 13: LCM of two numbers is always equal to their product when they are prime
True. LCM of two prime numbers is their product, as stated in Statement 7.
Statement 14: LCM of two numbers is always equal to their product when they are relatively prime
True. LCM of two relatively prime numbers is their product, as stated in Statement 9.
Statement 15: LCM of two numbers is always equal to their product when they are equal
True. LCM of two equal numbers is their product because they are already multiples of each other.
Statement 16: LCM of two numbers is always equal to their product when one of them is 1
True. LCM of 1 and any number is that number because 1 is a factor of any number.
Statement 17: LCM of two numbers is always equal to their product when they have no common factors other than 1
True. LCM of two numbers that have no common factors other than 1 is their product, as stated in Statement 3.
Statement 18: LCM of two numbers is always equal to their product when they are consecutive even numbers
False. LCM of two consecutive even numbers is equal to half of their product because one of them is always divisible by 2.
Statement 19: LCM of two numbers is always equal to their product when they are consecutive odd numbers
False. LCM of two consecutive odd numbers is equal to their product because they are already relatively prime.
Statement 20: LCM of two numbers is always equal to their product when one of them is a multiple of the other
True. LCM of two numbers where one is a multiple of the other is equal to the larger number because the larger number is already a multiple of the smaller number.
Conclusion
In conclusion, LCM is a useful mathematical concept that has many applications in different fields. We have explored some statements about LCM and determined which ones are true. It is important to understand the properties of LCM to use it effectively in problem-solving and analysis.
