Hcf & Lcm Questions for SSC GD

What is HCF (Highest Common Factor)?

HCF is the largest number that divides two or more numbers without leaving a remainder. Think of it as the biggest piece you can cut from multiple ropes without waste. In SSC GD, HCF questions often appear as 'distribute equally' or 'arrange in groups' problems. For example, if you have 24 apples and 36 oranges and want to pack them into identical boxes with no fruit left over, HCF tells you the maximum number of boxes.

• HCF is always smaller than or equal to the smallest number
• HCF of two coprime numbers (like 7 and 11) is always 1
• Prime factorization method: take the lowest power of common prime factors
• Division method (Euclidean): repeatedly divide until remainder is 0

What is LCM (Least Common Multiple)?

LCM is the smallest number that is a multiple of two or more given numbers. Imagine two bells ringing at different intervals—LCM tells you when they ring together again. In SSC GD, LCM appears in time-based problems: 'When will two workers meet again?' or 'When will three lights blink together?' LCM is always greater than or equal to the largest number.

• LCM is always greater than or equal to the largest number
• LCM of two coprime numbers is their product
• Prime factorization method: take the highest power of all prime factors
• Used in problems involving cycles, meetings, and repetition

Prime Factorization Method (Most Reliable)

Break each number into prime factors. For HCF, multiply the common factors with their lowest powers. For LCM, multiply all prime factors with their highest powers. This method works for any number of integers and is less error-prone than other methods in exam conditions.

• Step 1: Write prime factorization of each number
• Step 2: For HCF, circle common primes and use lowest power
• Step 3: For LCM, use all primes with highest power
• Example: 12 = 2² × 3, 18 = 2 × 3² → HCF = 2¹ × 3¹ = 6, LCM = 2² × 3² = 36

The Golden Rule: HCF × LCM = Product of Two Numbers

This relationship is a time-saver in SSC GD. If you know HCF and one number, you can find LCM without factorization. If you know LCM and HCF, you can verify your answer. This rule applies only to two numbers, not three or more.

• Formula: HCF(a,b) × LCM(a,b) = a × b
• Use this to check your work quickly
• Rearrange to find missing value: LCM = (a × b) / HCF
• Does NOT work for three or more numbers

Word Problem Recognition in SSC GD

SSC GD tests HCF and LCM through real-world scenarios. Learn to spot the keywords. HCF problems ask 'what is the maximum/largest/greatest?' LCM problems ask 'when will they meet again?' or 'what is the minimum/least/smallest?' Reading the question carefully is half the battle.

• HCF keywords: distribute equally, arrange in groups, maximum, largest, greatest, divide without remainder
• LCM keywords: meet again, ring together, repeat, minimum, least, smallest, common time
• Always identify what is being asked before solving
• Common SSC GD pattern: 'Bells ring at intervals of X and Y minutes. When will they ring together?'—this is LCM

• Confusing HCF and LCM: Remember HCF is smaller (High Common Factor = High... but smaller number), LCM is larger. Use the analogy: HCF is the 'common ground' (smaller), LCM is the 'common future' (larger).
• Forgetting to use lowest/highest power in prime factorization: Many aspirants write all prime factors for both HCF and LCM. HCF uses LOWEST power of common primes only; LCM uses HIGHEST power of ALL primes.
• Applying HCF × LCM = Product rule to three numbers: This rule works ONLY for two numbers. For three or more, use prime factorization only.
• Misreading word problems: A question asking 'arrange in equal groups' is HCF, not LCM. Slow down and underline the key verb.
• Arithmetic errors in prime factorization: Double-check your factorization. A single wrong prime factor ruins the entire answer. Verify by multiplying back.

• HCF = High Common Factor = Highest (but results in smaller number). LCM = Least Common Multiple = Lowest (but results in larger number). Think: HCF is 'exclusive' (fewer numbers divide it), LCM is 'inclusive' (more numbers divide it).
• HCF is the 'biggest piece you can cut equally.' LCM is the 'smallest time they meet again.' Use these two mental images for every problem.
• For prime factorization: HCF = Common primes with SMALL powers. LCM = All primes with BIG powers. Small = HCF, Big = LCM.
• Keyword shortcut: 'Distribute/Arrange/Divide' = HCF. 'Meet/Ring/Repeat/Together' = LCM.

• SSC GD typically asks 1-2 HCF/LCM questions in the 60-minute math section. These are usually worth 2-4 marks and are considered easy-to-medium difficulty if you know the method.
• Recent papers show a shift toward word problems over pure numerical questions. Always read the full question and identify HCF vs LCM before calculating.
• Time management: HCF/LCM should take 2-3 minutes max per question. If you're stuck, use prime factorization (slowest but most reliable) rather than guessing.
• In SSC GD, calculators are NOT allowed. Practice mental math for small numbers (up to 100). For larger numbers, prime factorization is your safest bet.
• Common SSC GD trap: A question might give you HCF and ask for LCM, or vice versa. Use the HCF × LCM = Product rule to cross-check your answer in 10 seconds.