CAT HCF and LCM of two or more numbers - Practice Questions & MCQ

Highest Common Factor (HCF):

The HCF of two or more numbers is the largest positive integer that divides them without leaving a remainder. It is also known as the Greatest Common Divisor (GCD). HCF is used to solve problems related to dividing objects into smaller equal groups.

Least Common Multiple (LCM):

The LCM of two or more numbers is the smallest multiple that is evenly divisible by all of the given numbers. LCM is useful when solving problems that involve finding the least number of repetitions required for multiple events to synchronise or repeat together.

Tips and Tricks for HCF and LCM:

Finding HCF: Euclidean Algorithm

To find the HCF of two numbers, use the Euclidean algorithm: 

- Divide the larger number by the smaller number. 

- If the remainder is zero, the smaller number is the HCF. 

- If the remainder is not zero, replace the larger number with the smaller number and the remainder as the new smaller number. 

- Repeat the process until the remainder becomes zero, then the last non-zero remainder is the HCF.

Finding LCM: Prime Factorization Method

To find the LCM of two or more numbers, use the prime factorization method: 

- Express each number as a product of prime factors. 

- Take the highest power of each prime factor that appears in any of the numbers. - Multiply all the selected prime factors together to obtain the LCM.

Example:

Question: Find the HCF and LCM of 18 and 24. 

Explanation: 

Step 1: Find the prime factors of 18: 18 = 2 × 3² 

Step 2: Find the prime factors of 24: 24 = 2³ × 3 

HCF: The common prime factors are 2 and 3, so the HCF is 2 × 3 = 6. 

LCM: The highest powers of prime factors in 18 and 24 are 2³ and 3². So, the LCM is 2³ × 3² = 72. Therefore, the HCF of 18 and 24 is 6, and the LCM is 72.

Solved Examples:

Question 1: Find the HCF and LCM of 15 and 25. 

Explanation: 

Step 1: Find the prime factors of 15: 15 = 3 × 5 

Step 2: Find the prime factors of 25: 25 = 5² HCF: 

The only common prime factor is 5, so the HCF is 5. 

LCM: The highest powers of prime factors in 15 and 25 are 3 and 5². So, the LCM is 3 × 5² = 75. Therefore, the HCF of 15 and 25 is 5, and the LCM is 75.

Question 2: Three numbers, A, B, and C, have HCF 5 and LCM 150. If A = 25, find B and C. 

Explanation: Since A = 25 and the HCF is 5, we can express B and C as B = 5x and C = 5y, where x and y are coprime. Given that the LCM of A, B, and C is 150, we can calculate the LCM using the HCF and individual numbers. 

LCM(A, B, C) = (HCF × A × B × C) / (5 × 25) = 150 

Simplifying, we get B × C = 375. 

To find two numbers whose product is 375, we can use trial and error. 

In this case, B = 15 and C = 25 satisfy the conditions. 

Therefore, B = 15 and C = 25.