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In mathematics, understanding the concepts of LCM (Least Common Multiple) and HCF (Highest Common Factor) is essential for solving a wide range of problems involving numbers. These concepts help us find common ground between numbers, whether it is the smallest number that two or more numbers divide into evenly (LCM) or the largest number that divides two or more numbers exactly (HCF).
LCM and HCF are closely linked to other important topics such as divisibility rules, prime factorization, and simplification of fractions. Mastering these concepts will not only help you in competitive exams but also in real-life situations like scheduling events, sharing resources, and simplifying calculations.
Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers. In other words, it is the smallest number into which all the numbers divide without leaving a remainder.
Highest Common Factor (HCF), also known as Greatest Common Divisor (GCD), of two or more numbers is the largest number that divides all the given numbers exactly without leaving a remainder.
To find LCM and HCF efficiently, we use prime factorization, which means expressing a number as a product of prime numbers.
From the factor trees above:
The HCF is the product of the prime factors common to both numbers, taken with the smallest powers:
HCF = \(2^{\min(3,2)} \times 3^{\min(1,2)} = 2^2 \times 3^1 = 4 \times 3 = 12\)
The LCM is the product of all prime factors present in either number, taken with the highest powers:
LCM = \(2^{\max(3,2)} \times 3^{\max(1,2)} = 2^3 \times 3^2 = 8 \times 9 = 72\)
There are three common methods to find LCM and HCF:
graph TD A[Start] --> B{Choose method} B --> C[Listing multiples/factors] B --> D[Prime factorization] B --> E[Division method] C --> C1[Write multiples/factors] C1 --> C2[Find common multiples/factors] C2 --> C3[Select smallest common multiple for LCM] C2 --> C4[Select greatest common factor for HCF] D --> D1[Find prime factors of each number] D1 --> D2[Use min exponents for HCF] D1 --> D3[Use max exponents for LCM] E --> E1[Divide numbers by common prime factors] E1 --> E2[Multiply divisors for HCF] E1 --> E3[Multiply divisors and remaining numbers for LCM]For any two numbers \(a\) and \(b\), there is an important relationship:
This means the product of the LCM and HCF of two numbers equals the product of the numbers themselves. This relationship is very useful for quickly finding one value if the other two are known.
| Numbers (a, b) | LCM(a,b) | HCF(a,b) | Product a x b | LCM x HCF |
|---|---|---|---|---|
| 12, 18 | 36 | 6 | 216 | 216 |
| 24, 36 | 72 | 12 | 864 | 864 |
| 8, 20 | 40 | 4 | 160 | 160 |
Step 1: Find prime factors of each number.
24 = \(2^3 \times 3\)
36 = \(2^2 \times 3^2\)
Step 2: For HCF, take the minimum powers of common primes.
HCF = \(2^{\min(3,2)} \times 3^{\min(1,2)} = 2^2 \times 3 = 4 \times 3 = 12\)
Step 3: For LCM, take the maximum powers of all primes.
LCM = \(2^{\max(3,2)} \times 3^{\max(1,2)} = 2^3 \times 3^2 = 8 \times 9 = 72\)
Answer: HCF = 12, LCM = 72
Step 1: Write the numbers side by side and divide by common prime factors.
| Divisor | 48 | 180 |
|---|---|---|
| 2 | 24 | 90 |
| 2 | 12 | 45 |
| 3 | 4 | 15 |
| 3 | 4 | 5 |
| 4 | 5 |
Step 2: Multiply the divisors to get HCF.
HCF = \(2 \times 2 \times 3 \times 3 = 36\)
Step 3: Multiply HCF by the remaining numbers to get LCM.
LCM = \(36 \times 4 \times 5 = 720\)
Answer: HCF = 36, LCM = 720
Step 1: Identify the problem as finding the LCM of 12 and 15.
Step 2: Prime factorization:
12 = \(2^2 \times 3\)
15 = \(3 \times 5\)
Step 3: LCM = \(2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60\)
Answer: Both events will coincide after 60 days.
Step 1: Prime factorize each number.
18 = \(2 \times 3^2\)
24 = \(2^3 \times 3\)
30 = \(2 \times 3 \times 5\)
Step 2: For HCF, take the minimum powers of common primes.
Common prime factors are 2 and 3.
Minimum power of 2 = \(2^1\)
Minimum power of 3 = \(3^1\)
HCF = \(2^1 \times 3^1 = 6\)
Step 3: For LCM, take the maximum powers of all primes.
Maximum power of 2 = \(2^3\)
Maximum power of 3 = \(3^2\)
Include 5 from 30 as \(5^1\)
LCM = \(2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360\)
Answer: HCF = 6, LCM = 360
Step 1: Find HCF of numerator and denominator.
Prime factorization:
84 = \(2^2 \times 3 \times 7\)
126 = \(2 \times 3^2 \times 7\)
HCF = \(2^{\min(2,1)} \times 3^{\min(1,2)} \times 7^{\min(1,1)} = 2^1 \times 3^1 \times 7^1 = 42\)
Step 2: Divide numerator and denominator by HCF.
\(\frac{84}{126} = \frac{84 \div 42}{126 \div 42} = \frac{2}{3}\)
Answer: Simplified fraction is \(\frac{2}{3}\)
When to use: When numbers are large and listing multiples is time-consuming or error-prone.
When to use: When two values among LCM, HCF, and product are known.
When to use: To better understand the relationship between LCM and HCF visually.
When to use: When asked to simplify fractions quickly and accurately.
When to use: To quickly identify factors of numbers and reduce calculation time.
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