HCF and LCM Questions for SSC CGL

HCF — What It Is and How to Find It

HCF (also called GCD) is the largest number that divides two or more numbers without remainder. Think of it as the biggest common 'building block' shared by all numbers. For example, HCF of 12 and 18 is 6 because 6 is the largest number that goes into both. In SSC exams, you'll use prime factorization or Euclidean algorithm. Prime factorization is faster for small numbers; Euclidean is cleaner for large numbers and always appears in 'find HCF of two large numbers' questions.

• HCF is always less than or equal to the smallest number in the set
• HCF of coprime numbers (like 7 and 11) is always 1
• Prime factorization method: take the lowest power of each common prime factor
• Euclidean algorithm: repeatedly replace larger number with (larger mod smaller) until remainder is 0
• HCF is useful in problems about dividing items into equal groups or finding largest tile size

LCM — What It Is and How to Find It

LCM (Least Common Multiple) is the smallest number that is a multiple of two or more given numbers. Think of it as the smallest common 'container' that holds all numbers as factors. For example, LCM of 4 and 6 is 12 because 12 is the smallest number divisible by both 4 and 6. In SSC, LCM appears in problems about bells ringing together, pipes filling a tank, or finding when events repeat. Prime factorization is the standard method: take the highest power of each prime that appears.

• LCM is always greater than or equal to the largest number in the set
• LCM of coprime numbers is their product
• Prime factorization method: take the highest power of each prime factor present
• LCM is useful in problems about cycles, repetition, and synchronized events
• For two numbers: LCM × HCF = Product of the two numbers (golden rule)

The Golden Rule and Quick Checks

The relationship HCF(a, b) × LCM(a, b) = a × b is the most powerful shortcut in SSC exams. If you know HCF and one number, you can instantly find LCM without factorizing again. This saves 30-40 seconds per question. Always verify your answers using this rule — if your HCF and LCM don't satisfy it, you've made an error. This is especially useful in multi-step problems where you need both values.

• Always check: HCF × LCM = Product of the two numbers
• If HCF(a, b) = h, then both a and b are multiples of h
• If LCM(a, b) = l, then l is a multiple of both a and b
• For three or more numbers, find HCF/LCM of first two, then with the third
• In word problems, HCF = 'largest equal share', LCM = 'when they meet again'

Common Word Problem Patterns in SSC CGL

SSC loves disguising HCF and LCM in real-world scenarios. Bells ringing, soldiers in rows, tiles on a floor, pipes filling tanks — these are all HCF or LCM problems in disguise. The key is recognizing the pattern: if the problem asks 'when will they happen together again' or 'largest size of equal groups', it's LCM or HCF respectively. Reading the question carefully is half the battle.

• Bells/events repeating: use LCM to find when they coincide again
• Dividing into equal groups (tiles, soldiers, items): use HCF for the largest group size
• Remainder problems: often involve HCF or divisibility rules
• Ratio simplification: HCF is used to reduce ratios to lowest terms
• Always define what you're looking for before calculating

• Confusing HCF and LCM — remember: HCF is smaller (High Common Factor sounds big but it's the highest of the small factors), LCM is larger (Least Common Multiple is the least of the big multiples). Use the size rule: HCF ≤ smallest number, LCM ≥ largest number.
• Forgetting to take the correct power of primes — in HCF take the LOWEST power of common primes; in LCM take the HIGHEST power of ALL primes. Reversing this is a classic trap.
• Not using the HCF × LCM = a × b rule to verify — many students calculate both but don't check. This catches 80% of arithmetic errors.
• In word problems, misidentifying whether you need HCF or LCM — 'largest equal groups' = HCF, 'when they meet/ring together' = LCM. Read twice, calculate once.
• Using Euclidean algorithm incorrectly — remember it's HCF(a, b) = HCF(b, a mod b), not a mod b directly. The order matters.

• HCF = High Common Factor but it's the SMALLEST of the two numbers (or less). LCM = Least Common Multiple but it's the LARGEST of the two numbers (or more). Size rule: HCF shrinks, LCM grows.
• Prime factorization: HCF takes MINIMUM power, LCM takes MAXIMUM power. Min-Max rule.
• Word problem: 'Divide equally' or 'largest' = HCF. 'Together again' or 'when next' = LCM.
• Golden check: HCF × LCM = a × b. If this fails, your answer is wrong. Always verify.

• SSC CGL Tier 1 typically has 1 direct HCF/LCM question and 1 word problem variant. Total ~2 minutes if you're fast. Tier 2 may have 1-2 more in data interpretation or word problems.
• Prime factorization is faster than Euclidean for numbers under 100. Use Euclidean only for large numbers (3+ digits) or when factorization is tedious.
• In recent papers (2022-2024), LCM word problems (bells, pipes, soldiers) are more common than pure HCF. Practice these patterns.
• Always simplify before calculating — if asked for HCF of 24, 36, 48, factor out 12 first: HCF(2, 3, 4) = 1, so HCF = 12 × 1 = 12. This saves time.
• In multi-number problems (3+ numbers), find HCF/LCM of first two, then combine with the third. Don't try to factorize all three at once — error-prone.