HCF and LCM: Solving Real-World Problems

1. What are HCF and LCM?

Before we jump into methods, let's solidify the concepts.

• Highest Common Factor (HCF): The HCF (also known as the Greatest Common Divisor or GCD) of two or more numbers is the largest number that divides all of them without leaving a remainder.
• Least Common Multiple (LCM): The LCM of two or more numbers is the smallest number that is a multiple of all of them.

Let's take a simple example: 12 and 18.

• Factors of 12 are: {1, 2, 3, 4, 6, 12}

• Factors of 18 are: {1, 2, 3, 6, 9, 18}
  The common factors are {1, 2, 3, 6}. The highest among them is 6. So, HCF(12, 18) = 6.

• Multiples of 12 are: {12, 24, 36, 48, 60, 72, ...}

• Multiples of 18 are: {18, 36, 54, 72, ...}
  The common multiples are {36, 72, ...}. The least among them is 36. So, LCM(12, 18) = 36.

Notice a key pattern: the HCF is generally smaller than the given numbers, while the LCM is generally larger. This is a crucial first clue when analyzing word problems.

2. The Prime Factorization Method

Listing out all factors and multiples is slow and impractical for large numbers. The prime factorization method you learned in the last lesson is far more efficient and reliable.

Let's summarize the procedure as a clear algorithm:

To calculate HCF and LCM of a set of numbers:

1. Prime Factorize: Express each number as a product of its prime factors in exponential form (e.g., $36=2^2\times 3^2$).
2. Calculate HCF:
   • Identify all the common prime factors.
   • For each common factor, take its lowest power.
   • Multiply these together. If there are no common prime factors, the HCF is 1.
3. Calculate LCM:
   • List all prime factors that appear in any of the numbers.
   • For each prime factor, take its highest power.
   • Multiply these together.

Example: Find the HCF and LCM of 96, 36, and 18.

1. Prime Factorization:

   • $96=3\times 32=3^1\times 2^5$
   • $36=4\times 9=2^2\times 3^2$
   • $18=2\times 9=2^1\times 3^2$

2. HCF:

   • Common prime factors are 2 and 3.
   • Lowest power of 2 is $2^1$.
   • Lowest power of 3 is $3^1$.
   • HCF = $2^1\times 3^1=6$.

3. LCM:

   • All prime factors involved are 2 and 3.
   • Highest power of 2 is $2^5$.
   • Highest power of 3 is $3^2$.
   • LCM = $2^5\times 3^2=32\times 9=288$.

3. An Important Formula

For any two positive integers, a and b, there is a direct relationship between the numbers and their HCF and LCM. This formula is the basis for many exam questions.

The formula is:

Important: This relationship holds true only for two numbers. It does not work for three or more numbers.

4. Decoding Word Problems

This is the most critical part of the lesson. Your ability to quickly and correctly identify whether a problem requires HCF or LCM is a key exam skill.

Application 1: Bells Ringing Together (LCM)

The "bells ringing" or "traffic lights changing" problem is a classic application of LCM. You are looking for a future point in time—a multiple—when all events align.

This video directly tackles this type of problem and highlights the common confusion students face.

Application 2: Finding the Largest Measuring Unit (HCF)

Now let's look at a typical HCF problem. Imagine you need to measure several different lengths using a single measuring tape and you want the tape to be as long as possible to minimize the number of measurements.

Problem: Find the length of the largest plank which can be used to measure exactly the lengths 4 m 50 cm, 9 m 90 cm, and 16 m 20 cm. (From GeeksforGeeks, LINK)

1. Identification: The keyword is "largest". We need a plank that can measure all three lengths exactly, meaning its length must be a factor of all three lengths. To find the largest such plank, we need the Highest Common Factor.

2. Preparation: First, convert everything to the same unit.

   • 4 m 50 cm = 450 cm
   • 9 m 90 cm = 990 cm
   • 16 m 20 cm = 1620 cm

3. Calculation: Find the HCF of 450, 990, and 1620.

   • $450=45\times 10=(9\times 5)\times (2\times 5)=2\times 3^2\times 5^2$
   • $990=99\times 10=(9\times 11)\times (2\times 5)=2\times 3^2\times 5\times 11$
   • $1620=162\times 10=(2\times 81)\times (2\times 5)=2^2\times 3^4\times 5$
   • HCF = $2^1\times 3^2\times 5^1=2\times 9\times 5=90$.

4. Final Answer: The length of the largest plank is 90 cm.

Now try a problem yourself. Read the problem carefully and decide if it's an HCF or LCM problem before you start calculating.

Conclusion

Great job today! We have taken a crucial step in building your problem-solving arsenal for the CAT. You now have a systematic way to approach a whole category of questions that can seem confusing at first glance.

Key Takeaways:

• HCF is for "dividing" or "grouping": Use it when you need to find the largest common measure for a set of items (e.g., largest tile size, maximum capacity of a container). The answer will be less than or equal to the smallest number given.
• LCM is for "aligning cycles": Use it when you need to find when events with different intervals will occur simultaneously (e.g., bells ringing, runners meeting). The answer will be greater than or equal to the largest number given.
• Prime factorization is your most reliable tool for calculating HCF and LCM.
• For two numbers a and b, remember the golden rule: HCF x LCM = a x b.

In our next lesson, we will explore another core concept from Number Systems: Remainders. We'll learn about key theorems that help solve complex remainder problems efficiently. You'll see how today's topic of HCF connects to problems where we need to find the largest number that divides several others while leaving specific remainders.