Questions on LCM

Question 1: Find the LCM of 12, 18, and 24.

Answer:

• 12 = 2² × 3
• 18 = 2 × 3²
• 24 = 2³ × 3

Now, to find the LCM, we need to take the highest power of each prime factor that appears in any of the numbers:

• The highest power of 2 is 2³.
• The highest power of 3 is 3².

Multiplying these together gives us:

LCM = 2³ × 3² = 8 × 9 = 72

So, the LCM of 12, 18, and 24 is 72

Question 2: The LCM of the two numbers is 360, and their HCF is 24. If one of the numbers is 120, find the other number.

Answer:

We know that the product of the HCF and LCM of two numbers is equal to the product of the two numbers.

So, for two numbers a and b with HCF = 24 and LCM = 360:

HCF × LCM = a × b
24 × 360 = 120 × b [Given a = 120]
(24 × 360)/120 = b
24 × 3 = b
Thus, b = 72.

Question 3: A factory manufactures products in batches of 16, 24, and 32 units. What is the minimum number of units the factory needs to produce so that each batch can be formed exactly?

Answer:

To find the minimum number of units the factory needs to produce so that each batch size (16, 24, and 32) can be formed exactly, we need to find the least common multiple (LCM) of these batch sizes.

The prime factorization of each batch size is as follows:

• 16 = 2⁴
• 24 = 2³ × 3
• 32 = 2⁵

To find the LCM, we take the highest power of each prime factor that appears in any of the batch sizes:

• The highest power of 2 is 2⁵.
• The highest power of 3 is 3¹.

So, the LCM of 16, 24, and 32 is 2⁵ × 3¹ = 32 × 3 = 96.

HCF (Highest Common Factor) and LCM (Least Common Multiple) concepts are the foundation of many mathematical operations and are essential in solving complex problems. HCF and LCM problems challenge your ability to find the greatest common factor and the smallest common multiple of numbers, and they require both logical and mathematical skills. So get ready to exercise your brain as we delve into the world of HCF and LCM problems and explore the exciting ways they can be used to solve challenging aptitude questions!