HCF and LCM: Easy Yet Tricky Problems Explained

HCF and LCM: Easy Yet Tricky Problems Explained

HCF and LCM: Easy Yet Tricky Problems Explained Simply

Understanding HCF (Highest Common Factor) and LCM (Least Common Multiple) is essential for solving many mathematical problems efficiently. This article discusses the formulas, tricks, important methods, and examples to help you master these concepts.

What is HCF (Highest Common Factor)?

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more given numbers without leaving a remainder.

Methods to Find HCF

1. Division Method

This method uses successive divisions to find the HCF. For two numbers, N1 and N2:

  1. Divide the larger number by the smaller number.
  2. Use the remainder as the divisor for the next division.
  3. Repeat until the remainder is 0.
  4. The last divisor is the HCF.

2. Prime Factorization Method

Express each number as a product of prime numbers. The product of the least powers of all common prime factors gives the HCF.

Example: Find the HCF of 96, 144, and 240.

  • Prime factorization:
    • 96 = 25×312^5 \times 3^1
    • 144 = 24×322^4 \times 3^2
    • 240 = 24×31×512^4 \times 3^1 \times 5^1
  • HCF = 24×31=482^4 \times 3^1 = 48

Quick Tricks to Find HCF

Example 1: Find HCF of 12 and 16.

  • Difference = 16−12=416 – 12 = 4.
  • Check if 4 divides both numbers.
  • HCF = 4.

Example 2: Find HCF of 18 and 22.

  • Difference = 22−18=422 – 18 = 4.
  • Factors of 4: 2×2×12 \times 2 \times 1.
  • 18 and 22 are divisible by 2.
  • HCF = 2.

What is LCM (Least Common Multiple)?

The Least Common Multiple (LCM) is the smallest number that is exactly divisible by each of the given numbers.

Methods to Find LCM

Prime Factorization Method

After prime factorization of numbers, multiply all prime numbers in their highest powers.

Example: Find the LCM of 96, 144, and 240.

  • Prime factorization:
    • 96 = 25×312^5 \times 3^1
    • 144 = 24×322^4 \times 3^2
    • 240 = 24×31×512^4 \times 3^1 \times 5^1
  • LCM = 25×32×51=14402^5 \times 3^2 \times 5^1 = 1440.

Quick Tricks to Find LCM

Example 1: Find LCM of 2, 4, 8, and 16.

  • Largest number = 16.
  • Check if 16 is divisible by all numbers.
  • LCM = 16.

Example 2: Find LCM of 2, 3, 7, and 21.

  • Largest number = 21.
  • 21 is not divisible by 2. Multiply 21 by 2.
  • Check divisibility of 42 by all numbers.
  • LCM = 42.

HCF and LCM Formulas

  1. Relationship Between HCF and LCM: Product of two numbers=HCF×LCM\text{Product of two numbers} = \text{HCF} \times \text{LCM}
  2. Co-prime Numbers:
    • Two numbers are co-prime if their HCF = 1.
  3. HCF and LCM of Fractions:
    • HCF = HCF of numeratorsLCM of denominators\frac{\text{HCF of numerators}}{\text{LCM of denominators}}
    • LCM = LCM of numeratorsHCF of denominators\frac{\text{LCM of numerators}}{\text{HCF of denominators}}

Solved Examples

Example 1: Find HCF of 15, 25, and 35.

  • Difference = 25−15=1025 – 15 = 10.
  • Factors of 10: 5×2×15 \times 2 \times 1.
  • All numbers are divisible by 5.
  • HCF = 5.

Example 2: Find LCM of 12, 36, 60, and 108.

  • Largest number = 108.
  • Check divisibility of 108 by all numbers.
  • LCM = 108.

Conclusion

Mastering HCF and LCM is easier when you understand the logic and practice consistently. Use the tricks and methods outlined here to simplify your problem-solving process!

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