Question

The LCM and HCF of two numbers are equal, then the numbers must be (1) prime (2) co-prime (3) composite (4) equal

Short answer

Question: If the LCM and HCF of two numbers are equal, then these numbers must be ___________. Answer: equal

Step-by-step solution

Recalling the relationship between LCM and HCF

Let's recall the relationship between the LCM and HCF of two numbers. For any two numbers 'a' and 'b', their product is equal to the product of their LCM and HCF. Mathematically, this can be represented as: a * b = LCM(a, b) * HCF(a, b) Given that LCM(a, b) = HCF(a, b), let's call this value 'k'. So in this case, the relationship becomes: a * b = k * k Now, let's analyze the properties of prime, co-prime, composite, and equal numbers.

Prime Numbers

Prime numbers are those numbers greater than 1 that have only 1 and itself as their factors. If both 'a' and 'b' are prime, their HCF will be 1. However, in our case, the HCF is equal to the LCM, so the numbers cannot both be prime. Thus, option (1) is ruled out.

Co-prime Numbers

Co-prime numbers are those numbers whose HCF is 1. As in the prime numbers' case, since our HCF is equal to the LCM, the numbers cannot be co-prime. Thus, option (2) is also ruled out.

Composite Numbers

Composite numbers are the numbers which have more factors than just 1 and themselves. However, having them both be composite doesn't guarantee that their LCM will be equal to their HCF. For example, if a=6 (which is composite) and b=8 (also composite), the LCM(6,8)=24 while HCF(6,8)=2. Thus, option (3) also isn't necessarily true.

Equal Numbers

Let's now check if having both numbers 'a' and 'b' equal to each other will satisfy the given condition. If a=b, then the LCM of the two equal numbers is also the same number, and their HCF is also that same number. In this case, LCM(a, b) = HCF(a, b) = a (or b). This satisfies our initial condition from Step 1. Thus, option (4) is the correct choice. The two numbers must be equal for their LCM and HCF to be the same.

Key concepts

Prime Numbers

Prime numbers are fascinating because they serve as the building blocks of our number system. By definition, a prime number has exactly two distinct positive divisors: 1 and itself. This is why numbers like 2, 3, 5, 7, and 11 are considered prime.
Prime numbers have unique properties, particularly in multiplication and factorization. They cannot be divided exactly by any other number except 1 and themselves, which means their

• LCM (Least Common Multiple) is typically the product of the numbers if the primes are different.
• HCF (Highest Common Factor) is 1 because they do not share any common factors other than 1.

In the context of the original exercise, if both numbers are prime, their HCF cannot match their LCM if the primes are different. Therefore, the two numbers in question cannot both be prime if their LCM and HCF are equal.

Co-prime Numbers

Co-prime numbers, also known as relatively prime or mutually prime numbers, are numbers that share no common factors other than 1. This means that for co-prime numbers, the Highest Common Factor (HCF) is always 1. Examples include pairs like (8, 15) or (35, 64).
The concept of co-prime numbers is crucial in problems involving number theory and divisibility. Some key attributes of co-prime numbers include:

• They can result from any combination of prime or composite numbers, provided their HCF is 1.
• They are not necessarily prime themselves (e.g., 8 and 15 are both composite, yet co-prime).

In the given exercise scenario, since the LCM and HCF must equal each other, this condition contradicts the nature of co-prime numbers. Thus, the two numbers cannot be co-prime.

Composite Numbers

Composite numbers are those that have more factors than just 1 and themselves. Essentially, they can be divided evenly by numbers other than 1 and the number itself. Some examples include 4, 6, 8, 9, and 12. Composite numbers are interesting because:

• They can be broken down into products of prime numbers, known as prime factorization.
• Their HCF with any other number can vary significantly based on the other number’s divisors.

In the context of understanding LCM and HCF relationships in exercises, simply being composite does not guarantee that the LCM will equal the HCF. Consider numbers like 6 (factors: 1, 2, 3, 6) and 8 (factors: 1, 2, 4, 8); despite both being composite, their LCM isn't equal to the HCF as shown earlier. Therefore, composite numbers are not the answer when the LCM and HCF are the same.

Equal Numbers

When considering number pairs whose LCM and HCF are equal, equal numbers easily fulfill this requirement. When two numbers are identical, say both are 'n', they share all factors, making both their LCM and HCF equal to the number itself. Some consequences of having equal numbers include:

• The LCM of two equal numbers is the number itself because the highest multiple they both share is simply the number itself.
• The HCF for equal numbers is the number too, as it's the greatest factor they both divide into evenly.

In exercises that ask for circumstances where the LCM equals the HCF, equal numbers like (3, 3) or (10, 10) satisfy this condition perfectly. Thus, when LCM and HCF are to be equal, the best candidates are indeed equal numbers.