Real-Life Applications of HCF and LCM

Least Common Multiple (LCM) is the smallest number that can be evenly divided by two or more given numbers. Whereas Highest Common Factor (HCF) is the largest number that can evenly divide two or more given numbers.

LCM and HCF can be easily found using two main methods:

• Division Method
• Prime Factorization Method

In this article, we will learn about HCF and LCM Definition, application of HCF and LCM and others in detail.

HCF and LCM

HCF(Highest Common Factor) is the highest number that exactly divides all the number whose HCF it is. It is denoted by HCF(a, b), where “a” and “b” are the numbers for which we want to find the highest common factor. For example, if you have numbers like 2, 4, and 6, the smallest number that all of them can divide into is 12.

LCM(Lowest Common Multiple) of two or more numbers is smallest number that is multiple of both the numbers. It is denoted by LCM(a, b), where “a” and “b” are the numbers for which we want to find the least common multiple. For example, if you have numbers like 6 and 30, the largest number they can both be divided by is 6.

Applications of HCF

Some applications of Highest Common Factor (HCF) are:

• Cooking Portions: When you’re cooking for a group of people and need to divide the ingredients into equal portions, you might use the HCF to determine the largest number of people you can serve without any leftovers.

• Gardening and Landscaping: If you’re planning a garden or arranging plants in a certain pattern in your backyard, knowing the HCF can help you determine the optimal spacing between plants for efficient use of space.

• Packing Boxes: When packing items into boxes for shipping or storage, you might want to maximize the space in each box while ensuring that all items fit snugly without wasted space. The HCF can help you determine the dimensions of the boxes to achieve this.

• Fencing a Field: If you’re building a fence around a field or garden, you might want to evenly space the fence posts along the perimeter. Knowing the HCF can help you determine the maximum distance between each post to ensure the fence is sturdy.

• Crafting Projects: In crafting or DIY projects that involve cutting materials like fabric, wood, or paper into smaller pieces, knowing the HCF can help minimize waste and optimize the use of materials.

• Music Beats and Rhythms: In music, rhythms are often composed of patterns that repeat at regular intervals. Understanding the HCF can help musicians and composers create complex rhythms with different note durations that align harmoniously.

• Tile Layout: When tiling floors or walls with tiles of different sizes, knowing the HCF of the tile dimensions can help you arrange them in a pattern that minimizes cutting and waste while achieving a visually pleasing layout.

• Budget Planning: In personal finance, understanding the HCF can help you budget your expenses by identifying common factors among different categories of spending. This can help you prioritize expenses and allocate your budget more effectively.

Applications of LCM

Some applications of Lowest Common Multiple (LCM) are:

• Baking and Cooking: When you’re adjusting recipes to make larger batches of food, knowing the LCM helps you scale up the ingredients while maintaining the correct proportions.

• Scheduling Events: If you’re planning events or meetings that need to occur regularly but not necessarily at the same time, finding the LCM of the intervals between occurrences helps you schedule them efficiently without conflicts.

• Bus or Train Timetables: Public transportation systems often operate on schedules with multiple routes. Finding the LCM of the intervals between buses or trains helps coordinate transfers and optimize travel times for passengers.

• Repeating Patterns in Art or Design: Artists and designers often use repeating patterns in their work. Knowing the LCM helps them create patterns that repeat seamlessly without gaps or overlaps.

• Purchasing Supplies: When buying supplies in bulk, such as office supplies or groceries, knowing the LCM of the quantities you need ensures that you purchase the right amount to last until the next restocking without excess waste.

• Fitness and Exercise: In interval training or circuit workouts, exercises are often performed in repeating cycles. Understanding the LCM of the intervals between exercises helps design efficient workout routines.

• Traffic Light Sequences: Traffic lights at intersections operate on cycles, with each light changing at regular intervals. Finding the LCM of these intervals helps synchronize traffic flow and minimize congestion.

• Music Composition: In music composition, understanding the LCM of different note durations helps create complex rhythms and time signatures that flow smoothly and harmoniously.

Real-Life Example Of HCF and LCM

Various real-life examples of HCF and LCM are added below:

Buying Tiles For a Room

Aditya wants to tile the floor of his living room and bedroom with square tiles. The living room measures 30 feet by 40 feet, while the bedroom measures 60 feet by 80 feet.

• Living Room: Aditya wants to buy square tiles that perfectly fit the dimensions of his living room. He needs to find the size of the tiles that will evenly cover the floor without any gaps.

• Bedroom: Similarly, Aditya wants to buy square tiles for his bedroom, ensuring they fit the dimensions of the room perfectly without any leftover spaces.

Finding HCF: The HCF of the dimensions of both rooms (30 and 60) is 30. This means that the largest square tile that can evenly fit both the living room and the bedroom without any gaps is 30 feet by 30 feet.

Finding LCM: The LCM of the dimensions of both rooms (30 and 60) is 60. This means that if Aditya buys square tiles with dimensions of 60 feet by 60 feet, he’ll have enough to cover both the living room and the bedroom without any leftover tiles.

So, in this real-life scenario, the HCF helps determine the largest tile size that can evenly fit both rooms, while the LCM helps calculate the total quantity needed to cover both rooms without any leftover tiles.

Relationship Between LCM and HCF

Product Relationship: If you multiply the LCM (Least Common Multiple) and the HCF (Highest Common Factor) of any two numbers together, you get the product of those two numbers. For example, if you have two numbers, let’s say 4 and 6, then LCM(4,6) × HCF(4,6) = 4 × 6.

Co-Prime Numbers: When you have co-prime numbers (numbers with no common factors other than 1), their HCF is always 1. So, for co-prime numbers, the LCM is equal to the product of the numbers themselves.

FAQs on Applications of LCM and HCF

What are practical applications of LCM?

Practical applications of LCM are Scheduling repetitive events, coordinating tasks, finding common multiples in project timelines.

What are practical applications of HCF?

Practical applications of LCM are Simplifying fractions, splitting items into equal groups, optimizing resources in various scenarios like gardening or packaging.

What is formula between HCF and LCM?

Relationship between their HCF and LCM is equal to the multiplication of the two given numbers, as shown by the equation HCF(m, n) × LCM (m, n) = m× n.