LCM and HCF Explained: Concepts, Formulas, and Easy Examples - Aptitude Questions & Answers
LCM and HCF Explained: Concepts, Formulas, and Easy Examples is one of the most important topics in Quantitative Aptitude. In this lesson, you will learn concepts, formulas, shortcuts, solved examples, and aptitude questions with answers. This topic is useful for exams like SSC, Bank, CAT, TCS, and other competitive exams.
LCM and HCF: Complete Guide with Formulas, Properties & Examples
What are LCM and HCF?
LCM (Least Common Multiple) and HCF (Highest Common Factor) are fundamental concepts in number theory that help in solving problems related to multiples, factors, divisibility, and fractions. These concepts are essential for competitive exams and mathematical problem-solving.
Key Difference: LCM is the smallest number that is a multiple of given numbers, while HCF is the largest number that divides given numbers exactly.
LCM - Least Common Multiple
Definition and Method
The LCM of two or more numbers is the smallest positive integer that is divisible by all the given numbers without leaving a remainder.
Prime Factorization Method (Step-by-Step):
- Find prime factors of each number
- Write each number as product of prime factors
- Take the highest power of each prime factor
- Multiply these highest powers to get LCM
Example Problem: Find LCM of 12 and 18
- Prime factors: 12 = 2² × 3¹, 18 = 2¹ × 3²
- Highest power of 2: 2² (from 12)
- Highest power of 3: 3² (from 18)
- LCM = 2² × 3² = 4 × 9 = 36
Example 2: Find LCM of 15, 25, and 30
- 15 = 3 × 5
- 25 = 5²
- 30 = 2 × 3 × 5
- Highest powers: 2¹, 3¹, 5²
- LCM = 2 × 3 × 25 = 150
HCF - Highest Common Factor
Definition and Method
The HCF (also called GCD - Greatest Common Divisor) is the largest number that divides all given numbers exactly without leaving a remainder.
Prime Factorization Method for HCF:
- Find prime factors of each number
- Identify common prime factors
- Take the lowest power of each common factor
- Multiply these to get HCF
Example Problem: Find HCF of 18 and 27
- Prime factors: 18 = 2 × 3², 27 = 3³
- Common factor: 3 only (2 is not common)
- Lowest power of 3: 3² (from 18)
- HCF = 3² = 9
Example 2: Find HCF of 36, 48, and 60
- 36 = 2² × 3²
- 48 = 2⁴ × 3¹
- 60 = 2² × 3¹ × 5¹
- Common factors: 2 and 3
- Lowest power of 2: 2²
- Lowest power of 3: 3¹
- HCF = 2² × 3 = 4 × 3 = 12
Important Properties of LCM and HCF
Property 1: Product Relationship
a × b = LCM(a, b) × HCF(a, b)
Example Problem: LCM and HCF of two numbers are 168 and 7. One number is 21. Find the other number.
- Let the numbers be a = 21 and b = ?
- Using formula: a × b = LCM × HCF
- 21 × b = 168 × 7
- 21 × b = 1176
- b = 1176 ÷ 21 = 56
Property 2: Ratio of Numbers
Numbers = aH and bH
LCM = H × a × b
Example Problem: Two numbers are in ratio 15:11, HCF is 13. Find LCM.
- Let numbers be 15H and 11H, where H = 13
- LCM = H × 15 × 11 = 13 × 165 = 2145
Property 3: LCM and HCF of Fractions
LCM of Fractions
Example: LCM of 4/5 and 3/7
LCM = LCM(4,3) ÷ HCF(5,7) = 12 ÷ 1 = 12
HCF of Fractions
Example: HCF of 4/5 and 3/7
HCF = HCF(4,3) ÷ LCM(5,7) = 1 ÷ 35 = 1/35
Property 4: For Three Numbers
HCF(a, b, c) = HCF(HCF(a, b), c)
Example: Find LCM and HCF of 12, 15, 20
LCM: LCM(12,15)=60, LCM(60,20)=60
HCF: HCF(12,15)=3, HCF(3,20)=1
Multiples and Common Multiples
Understanding Multiples
A multiple of a number is the product of that number with any integer (positive, negative, or zero).
Example: Multiples of 5
- Positive multiples: 5, 10, 15, 20, 25, ...
- Negative multiples: -5, -10, -15, -20, ...
- Zero is a multiple of every number (5 × 0 = 0)
Common Multiples
Numbers that are multiples of two or more given numbers are called common multiples.
Example: Common multiples of 4 and 6
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, ...
- Common multiples: 12, 24, 36, 48, ...
- LCM = 12 (smallest common multiple)
Types of LCM Problems
Type 1: Finding Least Number Divisible by Given Numbers
Example Problem: Find least number divisible by 10, 15, and 20
- Find LCM: 10=2×5, 15=3×5, 20=2²×5
- LCM = 2² × 3 × 5 = 60
- 60 is the smallest number divisible by all three
Type 2: Same Remainder When Divided
Example Problem: Find least number that when divided by 8, 10, 12 leaves remainder 4 each time
- Find LCM(8,10,12) = 120
- Required number = LCM + remainder = 120 + 4 = 124
Type 3: Different Remainders When Divided
Example Problem: Find least number that when divided by 4, 5, 6 leaves remainders 2, 3, 4 respectively
- Note: 4-2 = 5-3 = 6-4 = 2 (common difference)
- Find LCM(4,5,6) = 60
- Required number = LCM - common difference = 60 - 2 = 58
General Rule: If remainder difference is constant, number = LCM - constant
Highest Power of a Prime in Factorial
Legendre\'s Formula
To find the highest power of a prime p in n!, use:
until pᵏ > n
Example Problem: Highest power of 3 in 15!
- ⌊15/3⌋ = 5
- ⌊15/9⌋ = 1
- ⌊15/27⌋ = 0 (stop since 27 > 15)
- Total = 5 + 1 = 6
Example 2: Highest power of 12 in 50!
Solution: Since 12 = 2² × 3, find highest power of 2² and 3 separately
- Highest power of 2 in 50! = ⌊50/2⌋+⌊50/4⌋+⌊50/8⌋+⌊50/16⌋+⌊50/32⌋ = 25+12+6+3+1 = 47
- Highest power of 2² = ⌊47/2⌋ = 23
- Highest power of 3 in 50! = ⌊50/3⌋+⌊50/9⌋+⌊50/27⌋ = 16+5+1 = 22
- Since we need both factors, take minimum: min(23, 22) = 22
- Highest power of 12 = 22
Factors and Their Properties
What are Factors?
Factors of a number are integers that divide the number exactly without leaving a remainder.
Example: Factors of 18: 1, 2, 3, 6, 9, 18
Total Number of Factors
Total factors = (a+1) × (b+1) × (c+1) × ...
Example Problem: Total factors of 120
- 120 = 2³ × 3¹ × 5¹
- Total factors = (3+1) × (1+1) × (1+1) = 4 × 2 × 2 = 16
Sum of Factors
Sum of factors = (p⁰+p¹+...+pᵃ) × (q⁰+q¹+...+qᵇ) × (r⁰+r¹+...+rᶜ) × ...
Example Problem: Sum of factors of 480
- 480 = 2⁵ × 3¹ × 5¹
- Sum = (1+2+4+8+16+32) × (1+3) × (1+5)
- Sum = 63 × 4 × 6 = 1512
Product of Factors
Product of all factors = N^(F/2)
Example Problem: Product of factors of 162
- 162 = 2¹ × 3⁴
- Total factors F = (1+1)×(4+1) = 10
- Product = 162^(10/2) = 162⁵
Even Factors
Even factors = a × (b+1) × (c+1) × ...
If N is odd, even factors = 0
Example Problem: Even factors of 120
- 120 = 2³ × 3¹ × 5¹
- Even factors = 3 × (1+1) × (1+1) = 3 × 2 × 2 = 12
LCM and HCF: Frequently Asked Questions
What\'s the relationship between LCM and HCF of two numbers?
The product of two numbers equals the product of their LCM and HCF. For numbers a and b: a × b = LCM(a,b) × HCF(a,b). This is a fundamental property used in many problems.
How to find LCM and HCF of more than two numbers?
For three numbers a, b, c: LCM(a,b,c) = LCM(LCM(a,b), c) and HCF(a,b,c) = HCF(HCF(a,b), c). Continue this process for more numbers.
What is the difference between factors and multiples?
Factors are numbers that divide a given number exactly (smaller or equal to the number). Multiples are obtained by multiplying the number by integers (larger or equal to the number).
How to quickly find highest power of a prime in factorial?
Use Legendre\'s formula: ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + ... until pᵏ > n. This gives the exponent of prime p in n!.
Can HCF be greater than LCM?
No, for two or more numbers, HCF is always less than or equal to LCM. For identical numbers, HCF = LCM = the number itself.
Practice Problems
Problem 1
Two numbers are in ratio 5:6. Their HCF is 4. Find their LCM.
Problem 2
Find the least number which when divided by 12, 16, 18 leaves remainder 5 each time.
Problem 3
Highest power of 5 in 100! is?
Problem 4
Total number of factors of 360 is?
Problem 5
Find LCM and HCF of 2/3, 4/5, and 6/7.
Quick Reference Formulas
• a × b = LCM(a,b) × HCF(a,b)
• LCM of fractions = LCM(Num)/HCF(Den)
• HCF of fractions = HCF(Num)/LCM(Den)
• Total factors = (a+1)(b+1)(c+1)...
• Sum of factors = (p⁰+...+pᵃ)(q⁰+...+qᵇ)...
• Product of factors = N^(F/2)
• Highest power of p in n! = Σ⌊n/pᵏ⌋
• If N=2ᵃ×pᵇ×qᶜ..., Even factors = a×(b+1)×(c+1)...
Frequently Asked Questions
What is LCM and HCF Explained: Concepts, Formulas, and Easy Examples?
LCM and HCF Explained: Concepts, Formulas, and Easy Examples is an important aptitude topic used in competitive exams that tests your logical reasoning and problem-solving abilities.
Is LCM and HCF Explained: Concepts, Formulas, and Easy Examples important for competitive exams?
Yes, LCM and HCF Explained: Concepts, Formulas, and Easy Examples is frequently asked in SSC, Bank, CAT, TCS, and other placement exams. It's essential to master this topic for better scores.
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