What is the definition of HCF (Highest Common Factor)?

The HCF of two or more numbers is the largest number that divides all of them without leaving a remainder. For example, the HCF of 12 and 18 is 6.

What is the definition of LCM (Lowest Common Multiple)?

The LCM of two or more numbers is the smallest number that is a multiple of all of them. For example, the LCM of 4 and 5 is 20.

What is the relationship between HCF and LCM of two numbers a and b?

The relationship can be expressed as: HCF(a, b) × LCM(a, b) = a × b. This means that the product of the HCF and LCM of two numbers is equal to the product of the numbers themselves.

How do you find the HCF of 24 and 36 using prime factorization?

First, find the prime factorization: 24 = 2³ × 3¹ and 36 = 2² × 3². Then, take the lowest power of each common prime factor: HCF = 2² × 3¹ = 4 × 3 = 12.

Find the LCM of 6 and 8 using the prime factorization method.

Hint: Start by determining the prime factors of each number.

The prime factorization of 6 is 2¹ × 3¹, and for 8 it is 2³. To find the LCM, take the highest power of each prime: LCM = 2³ × 3¹ = 8 × 3 = 24.

What is a quick method to find HCF of two numbers?

A quick method is to use the Euclidean algorithm: repeatedly subtract the smaller number from the larger one until the two numbers are equal. This common value is the HCF.

If the HCF of two numbers is 12 and their LCM is 120, what are the two numbers?

Hint: Use the relationship HCF × LCM = a × b.

Let the two numbers be a and b. From the relationship: 12 × 120 = a × b. Thus, a × b = 1440. Possible pairs could be (12, 120), (24, 60), etc.

What is the LCM of 9 and 28?

Hint: Check if the numbers are coprime.

Since 9 (3²) and 28 (2² × 7¹) have no common factors, LCM = 9 × 28 = 252.

Explain how to find the HCF of three numbers: 32, 48, and 64.

Hint: Use prime factorization for each number.

The prime factorizations are: 32 = 2⁵, 48 = 2⁴ × 3¹, and 64 = 2⁶. The lowest power of common prime factors is 2⁴, so HCF = 16.

How do you calculate the LCM of two numbers using the HCF?

Hint: Remember the formula that relates HCF and LCM.

Use the formula: LCM(a, b) = (a × b) / HCF(a, b). For example, if a = 12 and b = 15, and HCF is 3, then LCM = (12 × 15) / 3 = 60.

What is the importance of HCF and LCM in real-life applications?

HCF is useful in simplifying fractions, while LCM helps in solving problems involving synchronization of events, such as finding common time intervals.