LCM and HCF Explained

Let’s break down the concepts of LCM (Lowest Common Multiple) and HCF (Highest Common Factor) in a simple way, along with tips, tricks, and examples to help you understand and solve problems quickly.

Important Terms Simplified

• Factors: A number that divides another number exactly.
  Example: Factors of 12 are 1, 2, 3, 4, 6, 12.
• Multiple: A number that is the product of another number and an integer.
  Example: Multiples of 3 are 3, 6, 9, 12, 15, ...
• Common Multiple: A number that is a multiple of two or more numbers.
  Example: Common multiples of 2 and 3 are 6, 12, 18, ...
• HCF (Highest Common Factor): The largest number that divides two or more numbers exactly.
  Example: HCF of 12 and 18 is 6.
• LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers.
  Example: LCM of 4 and 6 is 12.

Tips and Tricks

How to Find HCF and LCM

Method to Find HCF

• Prime Factorization Method: Break down each number into its prime factors. Multiply the common prime factors with the lowest powers.
  Example: 12 = 2² × 3 18 = 2 × 3² HCF = 2 × 3 = 6.
• Division Method: Divide the larger number by the smaller number. Replace the larger number with the smaller number and the smaller number with the remainder. Repeat until the remainder is 0. The last divisor is the HCF.
  Example: HCF of 12 and 18: 18 ÷ 12 = 1 (remainder 6) 12 ÷ 6 = 2 (remainder 0) So, HCF = 6.

Method to Find LCM

• Prime Factorization Method: Break down each number into its prime factors. Multiply the highest powers of all prime factors.
  Example: 12 = 2² × 3 18 = 2 × 3² LCM = 2² × 3² = 4 × 9 = 36.
• Division Method: Write the numbers in a row. Divide by the smallest prime number that divides at least one number. Repeat until no more division is possible. Multiply all divisors and remaining numbers to get LCM.
  Example: LCM of 12 and 18: 2 | 12, 18 3 | 6, 9 2, 3 LCM = 2 × 3 × 2 × 3 = 36.

Examples to Practice

Quick Tricks to Solve Problems Faster

• For HCF, look for the largest number that divides all given numbers.
• For LCM, look for the smallest number that is a multiple of all given numbers.
• For Fractions, remember the formulas: HCF = HCF of Numerators ÷ LCM of Denominators LCM = LCM of Numerators ÷ HCF of Denominators.
• Use the Product Rule: HCF × LCM = Product of Numbers (for two numbers only).

Final Tips

• Practice prime factorization for faster calculations.
• Use the division method for HCF to save time.
• Memorize the formulas for fractions to solve problems quickly.

Practice MCQs