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LCM and HCF

BSF Constable · Quantitative Aptitude

Also prepare for:CRPF Const CISF Const ITBP Const UP Const

Divisibility Rules →

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Core Concept

Read this first — the foundation of the topic

→Core Concept

HCF is the largest number that divides all given numbers completely. LCM is the smallest number that is divisible by all given numbers. Think of HCF as the biggest common piece, and LCM as the smallest common whole

💡Key Properties

• For two numbers a and b: HCF × LCM = a × b • HCF is always less than or equal to the smallest number • LCM is always greater than or equal to the largest number • If two numbers are co-prime (HCF = 1), then LCM = product of numbers

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Formula Block

Memorise — at least one formula appears in every paper

HCF × LCM = Product of two numbers

For three numbers: HCF(a,b,c) × LCM(a,b,c) ≠ a × b × c

HCF by Division Method: Keep dividing larger by smaller until remainder is zero

LCM = (a × b) ÷ HCF(a,b)

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Exam Patterns

What examiners ask — read before attempting PYQs

📋SSC CGL typically asks

direct HCF/LCM calculation, word problems involving bells/traffic lights, finding numbers when HCF/LCM is given, and ratio-based problems. Most questions are 2-mark difficulty level

⚡Powerful Shortcut - Product Rule

For any two numbers, if you know any three values among {number1, number2, HCF, LCM}, you can find the fourth instantly using: HCF × LCM = number1 × number2

✏️Worked Example 1

1

Prime factorization 48 = 2⁴ × 3¹ 72 = 2³ × 3²

2

HCF = product of lowest powers = 2³ × 3¹ = 8 × 3 = 24

3

LCM = product of highest powers = 2⁴ × 3² = 16 × 9 = 144 Verification: HCF × LCM = 24 × 144 = 3456 = 48 × 72

✏️Worked Example 2

1

Find LCM of 15 and 20 15 = 3 × 5 20 = 2² × 5

2

LCM = 2² × 3 × 5 = 60 minutes

3

Next time = 9:00 AM + 60 minutes = 10:00 AM Time-Saving Trick - Division Method: For HCF: Keep dividing larger number by smaller until remainder is zero. Last divisor is HCF. 48 ÷ 72: 72 = 48 × 1 + 24, then 48 = 24 × 2 + 0. HCF = 24

⚡Another Speed Trick - Co-prime Check

If two numbers have no common factors except 1, their HCF = 1 and LCM = their product

✏️Examples

(7,11), (15,16), (25,28)

⚠️#1 Most Common Mistake

Students confuse the formula application for three numbers. The golden rule HCF × LCM = product applies ONLY to two numbers. For three or more numbers, this formula does NOT work. Always work with pairs or use prime factorization method for multiple numbers.

This mistake costs students 2-4 marks per exam

💡Practical Tip

In word problems, LCM gives 'together again' scenarios (bells, traffic lights). HCF gives 'maximum equal groups' scenarios (distribution problems). Identify the scenario type first, then apply the appropriate concept.

🔑 Key Points

HCF × LCM = Product of two numbers (works only for exactly two numbers)
HCF is the largest common divisor, LCM is the smallest common multiple
For co-prime numbers: HCF = 1, LCM = product of the numbers
LCM = (a × b) ÷ HCF for any two numbers a and b
HCF ≤ smallest number, LCM ≥ largest number among given numbers
Prime factorization method: HCF uses lowest powers, LCM uses highest powers
Division method for HCF: divide larger by smaller repeatedly until remainder is zero
Word problems: LCM for 'together again', HCF for 'equal groups'
Three number formula: HCF × LCM ≠ a × b × c (common exam trap)
Speed check: if numbers share no common factors, they are co-prime

📌 Exam Facts

HCF of two consecutive numbers is always 1
HCF of two consecutive even numbers is always 2
LCM of fractions = LCM of numerators ÷ HCF of denominators
HCF of fractions = HCF of numerators ÷ LCM of denominators
For three numbers a, b, c: HCF(a,b,c) divides LCM(a,b,c)
Product of two numbers = HCF × LCM (valid only for exactly two numbers)
HCF is also called GCD (Greatest Common Divisor)
LCM is also called LCM (Least Common Multiple) or SCM (Smallest Common Multiple)

🚀 60-Second Revision

Formula: HCF × LCM = a × b (only for two numbers, not three)
Remember: Prime factorization uses lowest powers for HCF, highest for LCM
Trap: Three number product rule does NOT apply
Shortcut: Co-prime numbers have HCF = 1, LCM = their product
Pattern: LCM problems = 'when together again', HCF = 'maximum equal parts'
Quick check: HCF ≤ smaller number, LCM ≥ larger number
Speed tip: Use division method for HCF of two numbers

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Test yourself under real exam conditions

A timed BSF Constable mock shows exactly how LCM and HCF questions appear in the actual paper — and where you lose marks.

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