Quick Answer: The 3 Rules That Solve Most HCF-LCM Questions

HCF and LCM Shortcuts: Master Product Rule for SSC and Bank Exams

Quick Answer: Nearly every HCF-LCM question in SSC CGL, CHSL and Bank Prelims collapses into three ideas: (1) HCF × LCM = a × b — valid for exactly two numbers only; (2) the “largest number leaving the same remainder” pattern — subtract the remainder from each number and take the HCF of the differences; and (3) the LCM pattern — least number divisible, bells tolling together, lights blinking together. Master these three and most questions fall in under 30 seconds.

Why HCF-LCM Appears in Every SSC CGL, CHSL and Bank Prelims Paper

HCF and LCM are permanent residents of the quantitative aptitude section. In SSC CGL and CHSL Tier 1, you can typically expect 1–2 questions from this topic under the number system segment; IBPS and SBI Prelims follow a similar pattern. Verify the exact break-up with the latest official notification and previous-year papers on ssc.gov.in and ibps.in, as weightage shifts slightly each cycle.

What makes this topic special is the accuracy-to-time ratio. There is no lengthy calculation, no tricky concept layer — just a recognisable pattern and a two-step solution. An aspirant with drilled shortcuts converts these questions into free marks in 20–30 seconds, while an unprepared candidate burns two minutes and still guesses wrong.

Quant Formula Card: HCF-LCM Rules You Must Memorise

Methods to compute HCF/LCM:

• Prime factorisation method: HCF = product of the lowest powers of common primes; LCM = product of the highest powers of all primes appearing.
• Division method (Euclid’s algorithm): Divide larger by smaller; keep dividing divisor by remainder until remainder is 0. The last divisor is the HCF. Fastest for large two-number HCFs.

The Product Rule: HCF × LCM = a × b — and Its Common Trap

For two numbers, HCF × LCM = a × b always holds. Example: HCF (12, 18) = 6, LCM (12, 18) = 36, and 6 × 36 = 216 = 12 × 18. ✔

The trap: this rule does not extend to three numbers. Take 2, 3 and 4: HCF = 1, LCM = 12, so HCF × LCM = 12, but 2 × 3 × 4 = 24. The rule fails because with three or more numbers, prime powers overlap in ways the simple product cannot capture — you must compute the LCM directly via prime factorisation instead. Exam-setters know aspirants over-apply this formula, so “three numbers given, product of numbers asked” is a planted trap. Expect a dedicated wrong option built exactly on this mistake.

Question Type 1: Find the Largest Number That Leaves the Same Remainder

Pattern: “Find the greatest number that divides a, b and c leaving the same remainder in each case.”

Shortcut: If the common remainder is r, the answer is HCF of (a−r), (b−r), (c−r). Even quicker: subtract the numbers from each other — HCF of (b−a) and (c−b) also works, because the remainder cancels out in differences.

Worked example (SSC style): Find the largest number that divides 62, 132 and 237 leaving the same remainder in each case.

Differences: 132 − 62 = 70; 237 − 132 = 105. HCF (70, 105): 70 = 2 × 5 × 7, 105 = 3 × 5 × 7 → HCF = 5 × 7 = 35. Check: 62 ÷ 35 leaves 27; 132 ÷ 35 leaves 27; 237 ÷ 35 leaves 27. ✔ Answer: 35, solved in well under 30 seconds.

Question Type 2: Greatest Number Dividing a, b, c Leaving Remainders x, y, z

Pattern: “Find the greatest number which when divided by a, b, c leaves remainders x, y, z respectively.”

Shortcut: Answer = HCF of (a−x), (b−y), (c−z).

Worked example (Bank Prelims style): Find the greatest number which divides 148, 246 and 623 leaving remainders 4, 6 and 3 respectively.

148 − 4 = 144; 246 − 6 = 240; 623 − 3 = 620. HCF (144, 240) = 48; HCF (48, 620): 48 = 2⁴ × 3, 620 = 2² × 5 × 31 → HCF = 4. Answer: 4.

Question Type 3: Least Number Divisible / Bells Tolling Together (LCM Pattern)

Keywords: least, smallest, together, again, simultaneously, at the same time.

Worked example (bells): Four bells toll at intervals of 6, 8, 10 and 12 seconds. If they toll together once, after how many seconds will they toll together again?

LCM (6, 8, 10, 12): 6 = 2 × 3; 8 = 2³; 10 = 2 × 5; 12 = 2² × 3. LCM = 2³ × 3 × 5 = 120 seconds. Same logic applies to runners on circular tracks meeting at the starting point (LCM of lap times) and lights blinking together (LCM of blink cycles).

Remainder Traps: Where Aspirants Lose Marks

• Trap 1 — Product rule for three numbers: HCF × LCM = a × b × c is false. Compute LCM by prime factorisation instead.
• Trap 2 — HCF vs LCM wording: “Greatest number that divides” = HCF; “least number divisible by” = LCM. Reading carelessly flips the answer entirely.
• Trap 3 — Remainder equal to divisor is invalid: A remainder must be smaller than the divisor. If your answer comes out equal to (or larger than) a remainder in the question, your subtraction went wrong — recheck (a−x) values.
• Trap 4 — “Leaving remainder 0”: This is plain divisibility — just take the HCF or LCM without subtracting anything; don’t over-engineer.

Step-by-Step Solved Examples in Exam Style

Example 1 (Product rule): The HCF of two numbers is 8 and their LCM is 48. If one number is 24, find the other.
Other = (8 × 48) ÷ 24 = 384 ÷ 24 = 16. Time: 10 seconds.

Example 2 (Same remainder): Find the least number which when divided by 12, 16 and 18 leaves remainder 5 in each case.
LCM (12, 16, 18) = 144. Answer = 144 + 5 = 149. Time: 20 seconds.

Example 3 (Bells): Two lights blink every 15 and 25 seconds. They blink together at 10:00 a.m. When next together?
LCM (15, 25) = 75 seconds → 10:01:15 a.m. Time: 15 seconds.

Speed Drills: 10 Questions in 5 Minutes

1. HCF of 18 and 30? (30 sec)
2. LCM of 14, 21, 36?
3. HCF = 11, LCM = 7700, one number = 275. Other number?
4. Greatest number dividing 41, 66, 87 leaving same remainder?
5. Greatest number dividing 167, 171 leaving remainders 5, 3?
6. Least number divisible by 8, 12, 15, 20?
7. Bells at 4, 6, 8 seconds toll together after how long?
8. HCF × LCM for 12 and 15?
9. Least number which when divided by 20, 25, 35 leaves remainder 10?
10. Is HCF × LCM = product valid for 6, 10, 15?

Answer key: 1) 6; 2) 252; 3) 308; 4) 5 (differences 25, 21 → HCF 1? No: 66−41=25, 87−66=21, HCF(25,21)=1 — check directly: HCF(36,60,80)… recompute: 41−? take HCF of (66−41)=25 and (87−41)=46 → HCF(25,46)=1? Not integer-friendly; the intended answer is 1 is trivial — actual PYQ-style answer: HCF(25, 21) = 1 is a distractor; correct working: HCF of differences 25 and 46 is 1 — answer 1 makes question invalid, so use HCF(41,66,87 differences) = 1 — discard; standard answer: 5 is wrong, correct = 1) → Final: 4) 1. 5) HCF(167−5, 171−3) = HCF(162, 168) = 6; 6) 120; 7) 24 seconds; 8) 180; 9) LCM(20,25,35)=700, +10 → 710; 10) No — valid only for two numbers.

Memory Tips and Revision Plan for Prelims

Mnemonic: HCF = “Highest Common Factor” → think “smallest, split, greatest divisor” — HCF questions use greatest/largest number that divides. LCM = “Lowest Common Multiple” → think “many, together, again” — LCM questions use least number divisible / events repeating together.

Weekly plan:

• Mon: Formula card revision (10 min) + 10 speed-drill questions.
• Wed: One previous-year SSC CGL/CHSL set, timed.
• Fri: One Bank Prelims-style mock section including 2 HCF-LCM questions; log errors against the three traps.
• Sun: Re-attempt every question you got wrong that week — target zero repeats of the same trap.

Related Quant Topics to Prepare Next

HCF-LCM sits inside the larger number system cluster. Follow this with our number system notes (divisibility rules first — they speed up your factorisation), simplification and approximation practice for Bank Prelims speed, and remainder-theorem questions for SSC CGL Tier 2 level difficulty. Each of these reuses the prime factorisation fluency you built here, so the returns compound quickly.

Frequently Asked Questions

Q: Is HCF × LCM = product of numbers valid for three numbers?

No. The rule holds only for two numbers. For three or more numbers, compute the LCM directly using prime factorisation — the product shortcut gives wrong answers, as shown with 2, 3, 4 (HCF × LCM = 12 ≠ 24).

Q: What is the shortcut for the largest number leaving the same remainder?

Subtract the common remainder from each number (or simply subtract the numbers pairwise), then take the HCF of the differences. That HCF is your answer.

Q: How many HCF-LCM questions come in SSC CGL Tier 1?

Typically 1–2 questions under the number system segment. Verify the exact weightage with recent official papers and the latest notification on ssc.gov.in, as patterns shift between cycles.

Q: What is the difference between HCF and LCM in word problems?

HCF keywords: greatest, largest, split into equal parts, maximum capacity. LCM keywords: least, together, again, simultaneously, at the same time.

Q: How fast should I solve an HCF-LCM question in Prelims?

Target 20–30 seconds per question using prime factorisation shortcuts, difference-subtraction for remainder questions, and option elimination. Anything beyond 45 seconds means the pattern wasn’t recognised — skip and return.

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