SSC CGL Maths Number System: LCM, HCF

 HCF (Highest Common Factor) and LCM (Least Common Multiple) are among the most important topics in the SSC CGL Quantitative Aptitude section. These concepts form the foundation of many arithmetic and number-system questions and are frequently used in direct and application-based problems. 

A strong understanding of HCF and LCM helps candidates solve questions related to divisibility, ratios, remainders, word problems, and number properties with greater speed and accuracy. By learning the key formulas, relationships, and problem-solving techniques associated with HCF and LCM, aspirants can significantly improve their performance in the SSC CGL examination and strengthen their overall mathematical aptitude. 

HCF and LCM

HCF (Highest Common Factor) and LCM (Least Common Multiple) are foundational concepts in arithmetic. Preparing them is essential not only for direct problems but also for their broad application in various competitive exam questions. 

Basic Definition and Example

Consider two numbers, 12 and 18.

• The HCF (Highest Common Factor) is the number that is common to both numbers. For 12 and 18, the HCF is 6.

• The LCM (Least Common Multiple) is the smallest number that is a multiple of both numbers. For 12 and 18, the LCM is 36 (12 × 3 = 36, 18 × 2 = 36).

Relationship Between HCF, LCM, and Ratios

The LCM of two numbers can be expressed as HCF × (Ratio of First Number) × (Ratio of Second Number).

For numbers 12 and 18, extracting the common factor (HCF = 6) leaves 2 and 3. These (2 and 3) represent the ratios of the first and second numbers, respectively, which are co-prime. When you form a ratio of numbers, the common value extracted is always their HCF.

Verification: Given HCF = 6, Ratio_1 = 2, Ratio_2 = 3.

LCM = 6 × 2 × 3 = 36. This confirms the formula.

Product of Numbers and HCF/LCM Relationship

The product of the LCM and HCF of two numbers is equal to the product of the two numbers themselves.

LCM × HCF = First Number × Second Number

Verification:

Using the previous example of 12 and 18:

LCM = 36, HCF = 6

LCM × HCF = 36 × 6 = 216

First Number × Second Number = 12 × 18 = 216

The property is verified.

Throughout this chapter, it is crucial to understand these two fundamental concepts:

1. LCM = HCF × (Ratio of First Number) × (Ratio of Second Number)

2. LCM × HCF = First Number × Second Number

Method for Finding LCM: Prime Factorization

For larger numbers, the prime factorization method is effective for finding the LCM.

Steps:

1. Convert each number into its prime factors.

2. Identify all unique prime factors from all numbers.

3. For each prime factor, take the highest power it appears with across all numbers.

4. Multiply these highest powers together to get the LCM.

Example 1: Find the LCM of 12 and 18.

• Prime factorization:

• 12 = 2² × 3¹

• 18 = 2¹ × 3²

• Maximum powers:

• For prime factor 2: The maximum power is 2² (from 12).

• For prime factor 3: The maximum power is 3² (from 18).

• LCM = 2² × 3² = 4 × 9 = 36.

LCM Calculation Example with Three Numbers

Finding the LCM of three numbers follows the same principle as finding the LCM of two numbers. The most effective method is prime factorization, where each number is expressed as a product of its prime factors. After factorizing all the numbers, identify the highest power of each prime factor appearing in any of the numbers and multiply them together to obtain the LCM.  

Example 2: Find the LCM of 60, 24, and 32.

• Prime factorization:

• 60 = 2² × 3¹ × 5¹

• 24 = 2³ × 3¹

• 32 = 2⁵

• Maximum powers:

• For prime factor 2: The maximum power is 2⁵ (from 32).

• For prime factor 3: The maximum power is 3¹ (from 60 and 24).

• For prime factor 5: The maximum power is 5¹ (from 60).

• LCM = 2⁵ × 3¹ × 5¹ = 32 × 3 × 5 = 480.
  It is crucial to include all remaining unique prime factors from all numbers, even if they only appear in one number, to ensure it's the Least Common Multiple.

Methods for Finding HCF

There are several methods to find the Highest Common Factor (HCF) of two or more numbers, depending on the complexity of the problem. The most common approach is the factorization method, where the common factors of the given numbers are identified, and the greatest among them is selected as the HCF. For larger numbers, the difference method can be particularly useful, as it involves finding the differences between the numbers and determining their common factor. 

Method 1: Common Factor Extraction

For numbers like 60, 24, and 32, find common factors.

1. Extract 4 as a common factor: 60 = 4 × 15, 24 = 4 × 6, 32 = 4 × 8.

2. The remaining numbers (15, 6, 8) are not yet co-prime. Continue extracting common factors until the remaining numbers share no common factors other than 1.

3. In this case, 15, 6, and 8 have no further common factors. So, the common factor is 4.

Method 2: Difference Method (for larger numbers)

For larger numbers, find the HCF by examining the differences between the numbers. The HCF must be a factor of these differences.

Example: Find the HCF of 60, 24, and 32.

1. Calculate the differences between pairs of numbers:

• Difference between 60 and 24 = 36

• Difference between 32 and 24 = 8

2. Find the common factor of these differences (36 and 8).

3. The common factor of 36 and 8 is 4.

4. Therefore, the HCF of 60, 24, and 32 is 4.
   This method is particularly useful when dealing with large numbers.

Practice Problem 

Now that you have learned the key concepts, formulas, and methods related to HCF and LCM, it is important to apply them through practice questions. The following practice problems cover different question types, including finding HCF and LCM, determining numbers from given ratios, solving remainder-based questions, and applying important relationships between HCF and LCM. Work through these questions carefully to build confidence and enhance your problem-solving skills.

Question: Find the HCF of 35 and 30.

Solution using Factorization:

• 35 = 5 × 7

• 30 = 5 × 6
  The common factor is 5. Once 5 is taken out, the remaining factors (7 and 6) are co-prime, confirming 5 as the HCF.

Question: Two numbers are in the ratio 15:11. If their HCF is 13, what are the numbers?

Solution:

When the HCF is extracted to form a ratio, you can reverse this to find the original numbers.

• First Number = Ratio_1 × HCF = 15 × 13 = 195

• Second Number = Ratio_2 × HCF = 11 × 13 = 143
  The numbers are 195 and 143. 

Question: The ratio of two numbers is 6:5, and their HCF is 15. Find their LCM.

Solution:

Apply Concept 1: LCM = HCF × (Ratio_1) × (Ratio_2)

Given HCF = 15, Ratio_1 = 6, Ratio_2 = 5.

LCM = 15 × 6 × 5 = 15 × 30 = 450.

Without understanding the formula LCM = HCF × (Ratio_1) × (Ratio_2), solving such problems would be difficult.

Question: The ratio of two numbers is 8:5. Their LCM is 480. Find their HCF.

Solution:

Again, apply Concept 1: LCM = HCF × (Ratio_1) × (Ratio_2)

Given LCM = 480, Ratio_1 = 8, Ratio_2 = 5.

480 = HCF × 8 × 5

480 = HCF × 40

HCF = 480 / 40 = 12.

This formula will be repeatedly used and memorized in the classroom to ensure mastery.

Question: The HCF of two numbers is 5, and their LCM is 150. How many such pairs of numbers can be formed?

Solution:

Use the relationship: LCM = HCF × (Ratio_1) × (Ratio_2)

Given HCF = 5, LCM = 150. Let the ratios of the two numbers be 'a' and 'b'.

150 = 5 × a × b

a × b = 30

Now, find pairs of co-prime numbers (a, b) whose product is 30.

Pairs (a, b) such that a × b = 30 and (a, b) are co-prime:

1. (1, 30)

2. (5, 6)

3. (3, 10)

4. (2, 15)
   Therefore, 4 such pairs can be formed. 

Question: The HCF of two numbers is 5, and their LCM is 100. How many pairs of such numbers can be formed?

Solution:

Using the relation: LCM = HCF × (Ratio_1) × (Ratio_2)

100 = 5 × a × b

a × b = 20

Now, find pairs of co-prime numbers (a, b) whose product is 20.

Pairs (a, b) such that a × b = 20 and (a, b) are co-prime:

1. (1, 20)

2. (4, 5)
   The numbers in the ratio (a, b) must be co-prime to each other. (2, 10) is not a valid pair because 2 and 10 are not co-prime.Therefore, only 2 such pairs can be formed

Question: The HCF of two numbers is 12, and their LCM is 480. If one number is 96, find the other number.

Solution:

Apply Concept 2: LCM × HCF = First Number × Second Number

Given LCM = 480, HCF = 12, First Number = 96. Let the Second Number be 'N'.

480 × 12 = 96 × N

N = (480 × 12) / 96

N = 480 / 8

N = 60.

This problem directly applies Concept 2, which states that the product of the LCM and HCF is equal to the product of the two numbers.

Question: The LCM of two numbers is 12 times their HCF. The sum of their HCF and LCM is 403. If one number is 93, find the other number.

Solution:

1. Establish relationships:

• LCM = 12 × HCF

• LCM + HCF = 403

2. Solve for HCF and LCM:

• Substitute LCM from the first equation into the second: (12 × HCF) + HCF = 403

• 13 × HCF = 403

• HCF = 403 / 13 = 31.

• LCM = 12 × 31 = 372.

3. Find the second number:
   Use Concept 2: LCM × HCF = First Number × Second Number
   372 × 31 = 93 × N
   N = (372 × 31) / 93
   N = 372 / 3 (since 93 = 3 × 31)
   N = 124. 

Question: A shopkeeper has three containers of milk with capacities 345 liters, 120 liters, and 255 liters. Find the maximum size of a measuring vessel that can measure all three quantities exactly.

Understanding the Keyword: Words like maximum, greatest, longest, highest, biggest indicate that you need to find the HCF.

Solution Method 1: Factor Extraction

1. All numbers end in 5 or 0, so 5 is a common factor: 345 = 5 × 69, 120 = 5 × 24, 255 = 5 × 51.

2. Examine the remaining numbers (69, 24, 51). All are divisible by 3: 69 = 3 × 23, 24 = 3 × 8, 51 = 3 × 17.

3. The final remaining numbers (23, 8, 17) are co-prime.

4. The HCF is the product of the common factors extracted: 5 × 3 = 15.

Question: Three containers have capacities of 496 liters, 403 liters, and 713 liters. Find the biggest measure of a container that can measure all three quantities exactly.

Understanding the Keyword: Biggest measure implies finding the HCF.

Solution Method: Difference Method

1. Calculate the differences between pairs of numbers:

• Difference between 496 and 403 = 93

• Difference between 713 and 403 = 310

2. Find the HCF of these differences (93 and 310).

3. Both 93 and 310 are multiples of 31: 93 = 31 × 3, 310 = 31 × 10.

4. Since 3 and 10 are co-prime, 31 is the HCF of 93 and 310.

5. Therefore, the HCF of 496, 403, and 713 is 31.