LCM & HCF Tests — Pick a Level and Start
Tap a level to see every test in it. Start at Level 1 the day you finish the rules, then move to Level 2 with the timer running to push accuracy under pressure.
Level 1 Basic Checking… View tests
One rule applied cleanly — the LCM or HCF of a set of numbers, the product rule for two numbers, the next time three bells toll together, the greatest rod that measures three lengths, and the least number leaving the same remainder. Attempt these without the clock while the rules are still fresh.
Level 2 Tier I Standard Checking… View tests
Real SSC CGL Tier I difficulty — two ideas chained in one question: find the same-remainder divisor and then its HCF with a new number, recover an unknown exponent from a stated LCM, count the pairs with a given product and HCF, or work back from the least number to the multiplier. This is the score that tells you the chapter is exam-ready.
LCM and HCF is one of the shortest chapters in the SSC CGL Quantitative Aptitude paper and one of the most reliable for marks. Almost every question belongs to one of about a dozen patterns — bells tolling together, a greatest length that measures several lengths exactly, the least number leaving the same remainder, or two numbers given through their ratio — and once you know which of LCM or HCF a pattern wants, the arithmetic is short.
These practice tests put twenty-five LCM and HCF questions in front of you in one sitting, which is what turns the rules from something you can recite into something you apply without thinking. Every question comes with a full solution and the shortcut, in English or Hindi.
How Many LCM and HCF Questions Are Asked in SSC CGL?
The chapter usually gets one to two direct questions, and its methods reappear inside number system, simplification and several arithmetic word problems.
| Chapter | Tier I (out of 25) | Tier II Paper I (out of 30) |
|---|---|---|
| LCM & HCF | 1–2 | 1–2 |
| Number System | 2–3 | 3–4 |
| Decimal & Fraction | 1–2 | 1–2 |
| Simplification | 1–2 | 2–3 |
| Square & Cube Roots | 1 | 1–2 |
Typical spread across recent SSC CGL shifts. SSC does not publish a chapter-wise breakup, so treat these as planning ranges. Tier I maths is always 25 questions in a locked 15-minute section.
What You Will Practice in the LCM & HCF Chapter
Core topics
- Finding the HCF by prime factorisation and by Euclid’s division method
- Finding the LCM by prime factorisation and by the common-division ladder
- The product rule for two numbers, and why it fails for three
- LCM and HCF of fractions, mixed numbers and decimals
- Numbers written in prime-power form, and recovering an unknown exponent from a stated LCM or HCF
- Scaling — what happens to the LCM and HCF when both numbers are multiplied by the same factor
- Bells, traffic lights and runners on a circular track — the next simultaneous event, the k-th event, and how many in a period
- Least and greatest numbers divisible by a set, or leaving the same remainder, or with a constant gap between divisor and remainder
- Least number to add or subtract to reach a common multiple
- Greatest measuring rod, longest equal pieces, largest square tile and greatest number of identical packs
- The greatest divisor leaving a stated or unstated common remainder
- Numbers in a ratio, pairs from a given product or sum, and sums of reciprocals
- Least perfect square or cube divisible by a set, and counting multiples in a range
- HCF and LCM of algebraic expressions
Which one does the question want — LCM or HCF?
| The question says | You need | Typical wording |
|---|---|---|
| Greatest number that divides / measures / packs | HCF | Greatest rod measuring 1.8 m, 2.7 m and 4.5 m |
| Least number that is divisible by / fits all | LCM | Least number exactly divisible by 15, 18 and 25 |
| Repeating events happening together again | LCM | Bells toll every 4, 6 and 9 seconds |
| Same remainder from several numbers | HCF | Greatest number dividing 424, 984, 1684 with the same remainder |
| Same remainder on division by several divisors | LCM | Least number leaving remainder 5 on 6, 9 and 12 |
| Longest equal pieces with nothing wasted | HCF | Ropes of 132 cm and 176 cm cut into equal pieces |
The single fastest filter in this chapter: the word greatest almost always means HCF and the word least almost always means LCM — except in remainder-on-division questions, where the least number still comes from the LCM.
The four relations you must know cold
| Relation | What it says | Use it when |
|---|---|---|
| Product rule | For two numbers a and b: a × b = HCF × LCM | Any two of the four given, find the third |
| HCF divides LCM | The LCM is always a multiple of the HCF | “Which can / cannot be the HCF or LCM” |
| Sandwich | HCF ≤ smaller ≤ larger ≤ LCM | True or false statements |
| Co-prime split | With HCF h, the numbers are ha and hb with a, b co-prime, and LCM = hab | Ratio questions, “how many pairs”, “which pair” |
The product rule is for two numbers only. For 2, 2 and 2 the HCF is 2 and the LCM is 2, so HCF × LCM is 4 while the product is 8.
Remainder patterns — the three forms
| Condition | Form of the number | Example |
|---|---|---|
| Divisible by a, b, c | L × k | Least is L; greatest three-digit is 999 minus its remainder on L |
| Same remainder r on a, b, c | L × k + r | Remainder 5 on 6, 9, 12 → 36k + 5 → 41, 77, 113 … |
| Divisor minus remainder is the same d each time | L × k − d | Remainders 1, 3, 5 on 4, 6, 8 → 24k − 3 → 21, 45, 69 … |
Least number to subtract from N to reach a common multiple = the remainder of N on L; least to add = L minus that remainder. The two always sum to L.
Fractions and decimals
- LCM of fractions = LCM of the numerators ÷ HCF of the denominators. HCF of fractions = HCF of the numerators ÷ LCM of the denominators.
- Reduce every fraction first. 4/6, 6/9 and 10/15 are all 2/3, so their HCF is 2/3 — the rule applied without reducing gives 1/45.
- Mixed numbers become improper fractions before either rule is used.
- Decimals: pad to the same number of decimal places, drop the point, take the LCM or HCF of the whole numbers, then restore the same number of places. LCM(0.4, 0.06, 0.9) → 40, 6, 90 → 360 → 3.6.
Also worth memorising
- HCF(ka, kb) = k × HCF(a, b) and LCM(ka, kb) = k × LCM(a, b) — a common factor scales both
- HCF(a + b, b) = HCF(a, b) — the step that makes Euclid’s method work
- Least perfect square divisible by a set = LCM × the primes carrying odd powers; for a cube, raise each power to the next multiple of 3
- Numbers in the ratio a : b : c have HCF h and LCM h × LCM(a, b, c), so LCM ÷ HCF depends only on the ratio
- 1/x + 1/y = (x + y) ÷ (xy), and xy = HCF × LCM
- HCF(n, n + 1) = 1; HCF(n, n + 2) = 2 when n is even; LCM(n, n + 1) = n(n + 1)
Solved Examples — SSC CGL LCM & HCF
How the LCM & HCF Chapter Tests Work
- 25 questions per test, with a full question palette — answered, skipped, marked for review, not visited.
- Optional 30-minute timer — with the clock to build exam speed, without it while you are still learning the rules.
- Marking: +2 correct, −0.5 wrong, 0 unattempted — the same scheme as the real paper.
- Solutions, shortcut tricks and concept notes open after you submit, question by question.
- English and हिंदी, chosen at the start of every test.
- Free with a TrickySSC account — sign in once with Google or your mobile number, and every test is open to you.
How to Prepare LCM & HCF for SSC CGL
- Decide LCM or HCF before you write anything. Greatest that divides or measures is HCF; least that is divisible, or repeating events meeting again, is LCM. Getting this wrong costs the whole question.
- Learn Euclid’s method properly. For four-digit numbers it beats factorisation every time, and the same idea powers the difference trick in same-remainder questions.
- Memorise the product rule and its limit. a × b = HCF × LCM for two numbers, never for three.
- Convert units before you start. 3 m 60 cm is 360 cm, and 1 minute 20 seconds is 80 seconds. Mixed units are the commonest silent error in this chapter.
- Practise the co-prime split. Writing the numbers as ha and hb with a, b co-prime solves every “how many pairs”, “which ratio” and “largest difference” question in one move.
- Do Level 1 untimed, the same day you finish the rules. The aim is correct recall, not speed.
- Move to Level 2 with the 30-minute clock on. Level 2 chains two ideas per question, which is what a real Tier I shift does.
- Study it with number system and divisibility. Do number system and divisibility first — factorisation is the shared skill.
- Verify against a real paper. Once Level 2 is consistent, attempt a previous year paper and then a full mock test under locked sectional timing.
LCM & HCF Study Notes
Read the full chapter notes before you attempt Level 1 — every rule stated in letters, one worked example under each, the shortcuts and the seven traps SSC repeats.
SSC CGL LCM और HCF (महत्तम समापवर्तक और लघुत्तम समापवर्त्य) चैप्टर टेस्ट — हिंदी में
LCM और HCF SSC CGL गणित का सबसे छोटा और सबसे भरोसेमंद अंक देने वाला अध्याय है। लगभग हर शिफ्ट में इसका 1–2 प्रश्न सीधे आता है — जैसे घंटियों का दोबारा एक साथ बजना, कई लंबाइयों को ठीक-ठीक मापने वाली सबसे बड़ी छड़, एक ही शेषफल छोड़ने वाली सबसे छोटी संख्या, या अनुपात में दी गई दो संख्याएँ।
TrickySSC पर इस अध्याय के सभी टेस्ट हिंदी में उपलब्ध हैं — प्रश्न, विकल्प, विस्तृत हल और शॉर्टकट ट्रिक सब हिंदी में। दो स्तर हैं: लेवल 1 (आधारभूत) और लेवल 2 (टियर I स्तर)। प्रत्येक टेस्ट में 25 प्रश्न, वैकल्पिक 30 मिनट का टाइमर और +2 / −0.5 की वही मार्किंग जो असली परीक्षा में होती है।
याद रखने योग्य नियम: दो संख्याओं के लिए गुणनफल = HCF × LCM (तीन संख्याओं पर यह नियम नहीं चलता); “सबसे बड़ी संख्या जो बाँटे/मापे” = HCF और “सबसे छोटी संख्या जो विभाज्य हो” या “फिर कब एक साथ” = LCM; एक ही शेषफल बताया न हो तो अंतरों का HCF, बताया हो तो पहले घटाकर HCF; और भिन्नों का LCM = अंशों का LCM ÷ हरों का HCF।
पूरे अध्याय के नोट्स हिंदी में यहाँ पढ़िए: LCM और HCF स्टडी नोट्स।