CAT Relation between HCF, LCM and the two numbers - Practice Questions & MCQ

Introduction:

• HCF (Highest Common Factor) is the greatest number that divides two given numbers without leaving any remainder.
• LCM (Least Common Multiple) is the smallest number that is divisible by both the given numbers.

Understanding HCF:

HCF can be found by using the factorization method, where both numbers are expressed as a product of their prime factors.

For example, let's find the HCF of 24 and 36:

The prime factorization of 24 is 2 × 2 × 2 × 3 = 2³ × 3.

The prime factorization of 36 is 2 × 2 × 3 × 3 = 2² × 3².

Comparing the two factorizations, the common factors are 2 × 2 × 3 = 12.

Therefore, the HCF of 24 and 36 is 12.

Understanding LCM:

LCM can also be found using the factorization method, where both numbers are expressed as a product of their prime factors.

For example, let's find the LCM of 24 and 36:

The prime factorization of 24 is 2 × 2 × 2 × 3 = 2³ × 3.

The prime factorization of 36 is 2 × 2 × 3 × 3 = 2² × 3².

The LCM is obtained by taking the highest power of each prime factor that appears in the factorization.

Therefore, the LCM of 24 and 36 is 2³ × 3² = 72.

Relation between HCF and LCM:

HCF × LCM = Product of the two numbers

Using the previous example, we had HCF = 12 and LCM = 72. The product of the numbers 24 and 36 is 24 × 36 = 864.

So, 12 × 72 also equals 864. Hence, the relation is verified.

Tips and Tricks:

If two numbers are co-prime (i.e., their HCF is 1), then their LCM is simply the product of the numbers.

In other cases, use prime factorization to find the HCF and LCM of the numbers.

Solved Example:

Question: The product of two numbers is 4320 and their HCF is 12. Find their LCM.

Solution: We know that HCF × LCM = Product of the two numbers.

12 × LCM = 4320, so LCM = 4320/12 = 360.