If you’re preparing for a government exam like the JKSSB Social Forestry Worker test, you’ve definitely seen those LCM and HCF questions. They pop up every single time. I remember when I first started preparing, I thought, “I know this! It’s basic math!” But then I saw the tricky ways they were asked and realized I needed a much stronger grip. The good news? Once you truly understand these concepts, you can solve these problems in under a minute, saving precious time for other sections.

This guide is built from my own experience and years of teaching these concepts. We’ll walk through everything from the absolute basics to the clever shortcuts that exam setters love to test. My goal is to make this so clear and relatable that you’ll feel confident tackling any variation they throw at you.

Let’s Start at the Very Beginning: What Are LCM and HCF?

Before we jump into formulas, let’s get our foundations solid. Think of it like building a house—you need a strong base.

What’s a Factor?

Simply put, a factor is a number that divides another number completely, leaving no remainder. For example, the factors of 12 are the numbers that “fit” into it perfectly: 1, 2, 3, 4, 6, and 12.

What’s a Multiple?

A multiple is what you get when you multiply a number by any whole number. The multiples of 7 are 7, 14, 21, 28, and so on—it’s the 7 times table.

Highest Common Factor (HCF) or Greatest Common Divisor (GCD)

This is the largest number that can divide two or more given numbers without leaving a remainder. For example, the common factors of 18 and 24 are 1, 2, 3, and 6. The highest of these is 6. So, HCF(18, 24) = 6.

Least Common Multiple (LCM)

This is the smallest positive number that is a multiple of two or more given numbers. For example, some common multiples of 4 and 6 are 12, 24, 36… The least of these is 12. So, LCM(4, 6) = 12.

The Golden Rule You MUST Remember

For any two positive numbers, this relationship is your secret weapon:

HCF(a, b) × LCM(a, b) = a × b

This means if you know the HCF and one number, you can instantly find the LCM, and vice-versa. It’s a huge time-saver in exams.

How to Actually Find HCF and LCM: Choosing Your Method

There are a few ways to calculate these. Which one you use depends on the numbers you’re given.

Must-Know Properties and Shortcuts for Exams

These facts aren’t just theory; they’re your toolkit for eliminating wrong answers and solving problems at lightning speed.

• Size Matters: The HCF of numbers is always less than or equal to each number. The LCM is always greater than or equal to each number.
• Co-prime Numbers: If two numbers have an HCF of 1 (like 8 and 9), they are co-prime. Their LCM is simply their product (8 × 9 = 72).
• One Divides the Other: If one number is a multiple of the other (like 12 and 36), then the HCF is the smaller number (12) and the LCM is the larger number (36).
• Fractions Trick:
  • HCF of fractions = (HCF of Numerators) / (LCM of Denominators)
  • LCM of fractions = (LCM of Numerators) / (HCF of Denominators)

  Remember it as a “swap”: HCF uses LCM of the bottoms, LCM uses HCF of the bottoms.

• The Remainder Lifesaver: If a number leaves the same remainder ‘r’ when divided by several numbers, then: Number = [LCM of those divisors] × k + r. For the smallest such number, just use k = 1.

Walking Through Examples: From Simple to Smart

Let’s apply what we’ve learned. I’ll show you the step-by-step thinking process.

Example 1: The Straightforward One

Find HCF and LCM of 48 and 180.

Step 1: Prime Factorisation.
48 = 2⁴ × 3¹
180 = 2² × 3² × 5¹

Step 2: Find HCF. Take the lowest power of common primes: 2² and 3¹.
HCF = 2² × 3 = 4 × 3 = 12.

Step 3: Find LCM. Take the highest power of all primes: 2⁴, 3², 5¹.
LCM = 16 × 9 × 5 = 720.

Step 4: Verify with the Golden Rule. 12 × 720 = 8640. Is that equal to 48 × 180? Yes, it is! Perfect.

Example 2: The “Same Remainder” Classic

Find the smallest number which when divided by 6, 9, and 15 leaves a remainder of 4 each time.

Here’s the trick. If a number ‘N’ leaves remainder 4, then (N – 4) is perfectly divisible by 6, 9, and 15. So, (N – 4) must be a common multiple of these numbers. We want the smallest N, so we need the Least Common Multiple.

LCM(6, 9, 15) = 90. (6=2×3, 9=3², 15=3×5 → LCM = 2 × 3² × 5 = 90).
Therefore, N – 4 = 90 → N = 94.

Answer: 94. You can check it: 94 ÷ 6 gives remainder 4, 94 ÷ 9 gives remainder 4, and 94 ÷ 15 gives remainder 4.

Example 3: Working Backwards

The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.

This is where the Golden Rule shines. Let the other number be ‘x’.
HCF × LCM = Product of Numbers
12 × 180 = 36 × x
2160 = 36x
x = 2160 ÷ 36 = 60.

See how quick that was? No factorisation needed.

Practice Makes Perfect: Try These

Grab a pen and test yourself. I’ve included a mix of direct and application questions.

Set A: Quick Calculations

1. Find the HCF and LCM of 84 and 126.
2. What is the LCM of 15, 20, and 25?

Set B: Real-World Applications

3. Three bells ring at intervals of 6, 8, and 12 seconds. If they start together, after how many seconds will they next ring together?
4. What is the greatest 4-digit number that is exactly divisible by 12, 15, and 18?
5. Find the HCF of the fractions: ⁴⁄₉, ⁵⁄₁₂, and ⁷⁄₁₈.

Answers to Common Questions (FAQs)

Your Final Pre-Exam Checklist

Run through this mentally before your test:

• I know the definitions of factor, multiple, HCF, and LCM by heart.
• The formula HCF × LCM = Product of two numbers is second nature.
• I can confidently use both the Prime Factorisation and Division methods.
• I remember the “swap” rule for fractions: HCF uses LCM of denominators.
• For “same remainder” problems, I know to use: Number = LCM × k + remainder.
• I recognize special cases: co-prime numbers, and when one number is a multiple of another.
• I have a time goal: to solve any LCM/HCF question within 45-60 seconds.

If you’re comfortable with all of the above, you are more than ready to ace the quantitative aptitude section. These concepts are a cornerstone of your math preparation. Understand them deeply, practice consistently, and you’ll walk into that exam hall with confidence.

Wishing you the very best of luck in your preparations and your exam. You’ve got this!