HCF and LCM – Complete Short Notes for Competitive Exams

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1. Definitions

• HCF (Highest Common Factor)

  • The largest number that divides two or more numbers exactly.

  • Also called GCD (Greatest Common Divisor).

• LCM (Least Common Multiple)

  • The smallest number that is exactly divisible by two or more numbers.

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2. Properties of HCF & LCM

1. Relation between HCF and LCM (for two numbers (a) and (b))
   HCF(a,b)×LCM(a,b)=a×b

2. HCF of co-prime numbers = 1

3. LCM of co-prime numbers = product of numbers

4. HCF of a number and its multiple = smaller number

5. LCM of a number and its multiple = larger number

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3. Methods to Find HCF

A. Prime Factorization Method

• Express numbers as product of primes.

• HCF = product of common prime factors with smallest powers.

Example:
Find HCF of 48 and 60:
48=24×3,60=22×3×5

Common primes: $2^2$ and $3^1$ → HCF = $2^2 \times 3 = 12$

22×3=12)

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B. Division (Euclidean) Method

• Divide larger number by smaller number.

• Replace larger number by remainder.

• Repeat until remainder = 0 → last divisor = HCF

Example: HCF of 84 and 30:

1. (84 ÷ 30 = 2) remainder 24

2. (30 ÷ 24 = 1) remainder 6

3. (24 ÷ 6 = 4) remainder 0 → HCF = 6

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C. HCF by Listing Factors

• List all factors of numbers, pick largest common factor.

• Practical only for small numbers.

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4. Methods to Find LCM

A. Prime Factorization Method

• Express numbers as product of primes.

• LCM = product of all prime factors with highest powers.

Example:
48 = (2^4 × 3), 60 = (2^2 × 3 × 5)
LCM = (2^4 × 3 × 5 = 240)

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B. Using HCF

LCM(a,b)=HCF(a,b)a×b​

Example: a = 48, b = 60, HCF = 12

$$LCM = \frac{48 × 60}{12} = 240$$

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C. Listing Multiples

• List multiples of each number → first common multiple = LCM

• Practical only for small numbers

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5. HCF & LCM of More than Two Numbers

• HCF of three numbers:
  $$HCF(a,b,c)=HCF(HCF(a,b),c)$$

  • LCM of three numbers:

  $$LCM(a,b,c) = LCM(LCM(a,b), c)$$

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6. Tricks & Shortcuts

1. HCF of consecutive numbers = 1

2. LCM of consecutive numbers = product ÷ HCF = product ÷ 1 = product

3. HCF of all even numbers = at least 2

4. LCM of numbers with a common factor = multiply non-common factors × common factor

5. Use prime factorization for speed

6. Euclidean method faster for large numbers

7. Shortcut for 2-digit multiples:

   • HCF(36, 84) → Divide 84 by 36 → remainder 12 → Divide 36 by 12 → remainder 0 → HCF = 12

8. LCM of fractions:
   

   LCM(qp​,sr​)=HCF(q,s)LCM(p,r)​

9. HCF of fractions:
   HCF(qp​,sr​)=LCM(q,s)HCF(p,r)​

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7. Word Problems (Quick Tips)

1. HCF Problem:

   • “Divide a number into parts with maximum equal size → HCF of quantities”

2. LCM Problem:

   • “Events repeating in cycles → LCM of cycles”

3. Combined HCF & LCM Problem:

   • Use formula (HCF \times LCM = a \times b)

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8. Examples for Revision

1. Find HCF & LCM of 36 and 84:

   • Prime factorization: 36 = (2^2×3^2), 84 = (2^2×3×7)

   • HCF = (2^2×3 = 12), LCM = (2^2×3^2×7 = 252)

2. HCF of 48, 60, 72:

   • HCF(48,60)=12, HCF(12,72)=12 → HCF = 12

3. LCM of 8, 12, 20:

   • LCM(8,12)=24, LCM(24,20)=120 → LCM = 120

4. LCM of fractions:
   32​,65​ → LCM = $LCM(2,5)/HCF(3,6) = 10/3$

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9. One-Liner Formulas

• HCF × LCM = Product of numbers

• HCF(a,b,c) = HCF(HCF(a,b), c)

• LCM(a,b,c) = LCM(LCM(a,b), c)

• LCM of fractions = LCM(numerators)/HCF(denominators)

• HCF of fractions = HCF(numerators)/LCM(denominators)

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10. Exam Day Tips

• Always check if numbers are co-prime → speeds up HCF/LCM

• Use Euclidean method for large numbers instead of factorization

• Check for multiples in cycles (LCM) in word problems

• Memorize prime factorization tables up to 100 for speed

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✅ Quick Revision Checklist (High-Yield):

• HCF × LCM = Product of two numbers

• Use prime factorization for small numbers

• Use division method for large numbers

• Remember fraction formula for HCF & LCM

• Use LCM for repeating events and HCF for dividing objects equally