HCF and LCM are among the most reliable scorers in the CDS and OTA elementary mathematics paper. Almost every year you will see at least one direct question and one word problem built on them. This Cavalier guide explains both ideas in plain language, gives you the two standard methods, the must-know product rule, and fully solved examples so you can attempt these questions in under a minute.
Why HCF and LCM matter in CDS
Highest Common Factor (HCF) and Lowest Common Multiple (LCM) sit at the foundation of the number-system portion of the CDS and OTA syllabus. They look simple, yet examiners disguise them inside word problems about tiles, road junctions, bells and traffic lights so that candidates who only memorised a procedure get stuck.
The good news: once you understand what these two quantities actually represent, the questions become almost mechanical. A candidate who reads the story correctly can finish an HCF or LCM problem in well under a minute, and that saved time can be redirected to tougher topics like trigonometry or mensuration where each mark is harder to earn. Over a full paper, these few quick wins often decide who clears the cut-off.
Across decades of CDS papers the pattern barely changes: one straightforward computation question that tests whether you know the method, and one applied question dressed up as a real-life situation. The Cavalier approach is to drill both halves so that neither can surprise you on exam day.
HCF is about sharing or grouping into the largest equal parts. LCM is about events lining up or repeating together. Spot which one the story needs before you compute.
Factors and multiples: the building blocks
A factor of a number divides it exactly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. A multiple of a number is what you get on multiplying it by a whole number, so the multiples of 12 are 12, 24, 36, 48 and so on.
Every number has a finite list of factors but an unlimited list of multiples. This single difference explains the whole topic: HCF picks the biggest from the shared factors, so it is bounded above by the smallest number, while LCM picks the smallest from the shared multiples, so it is bounded below by the largest number.
Two numbers are called co-prime (or relatively prime) when the only factor they share is 1, such as 8 and 9, or 14 and 15. Co-prime pairs do not have to be prime themselves, and recognising them saves time because their HCF is always 1.
- 1 is a factor of every number.
- Every number is a factor and a multiple of itself.
- A prime number has exactly two factors: 1 and itself.
- Consecutive integers are always co-prime to each other.
What is the HCF?
The Highest Common Factor (also called the Greatest Common Divisor, GCD) of two or more numbers is the largest number that divides each of them exactly.
Take 18 and 24. Factors of 18 are 1, 2, 3, 6, 9, 18. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The common factors are 1, 2, 3 and 6, and the highest among them is 6. So HCF(18, 24) = 6. In practical terms, 6 is the largest size into which both an 18-unit and a 24-unit length can be cut without any leftover, which is exactly the kind of phrasing the exam likes to use.
The HCF can never be larger than the smallest of the given numbers. If one number is a factor of the other, the HCF is simply the smaller number, e.g. HCF(8, 24) = 8.
What is the LCM?
The Lowest Common Multiple of two or more numbers is the smallest number that is a multiple of each of them, that is, the smallest number that each of them divides exactly.
Take 4 and 6. Multiples of 4 are 4, 8, 12, 16, 20, 24… Multiples of 6 are 6, 12, 18, 24… The common multiples are 12, 24, 36… and the lowest is 12. So LCM(4, 6) = 12.
The LCM can never be smaller than the largest of the given numbers. If one number is a multiple of the other, the LCM is the larger number, e.g. LCM(5, 15) = 15.
Method 1: Prime factorisation
Express each number as a product of prime factors using powers. Then:
- HCF = product of common prime factors, each taken to the lowest power present.
- LCM = product of all prime factors that appear, each taken to the highest power present.
Consider 60 and 72.
Underline “lowest power for HCF, highest power for LCM”. Mixing these up is the single most common slip the examiner is hoping you make.
Method 2: The division (long-division) method
For HCF of two numbers, use the Euclidean idea: divide the larger by the smaller, then divide the previous divisor by the remainder, repeating until the remainder is 0. The last non-zero divisor is the HCF.
Find HCF(48, 36).
For LCM of several numbers, write them in a row and keep dividing by any prime that divides at least two of them, carrying down unchanged any number the prime does not divide. Stop when the bottom row is all co-prime, then multiply every divisor on the left by every number left in the bottom row.
Find LCM(8, 12, 20). Divide by 2 to get 4, 6, 10; divide by 2 again to get 2, 3, 5; these are co-prime. LCM = 2 × 2 × 2 × 3 × 5 = 120. The same ladder can be reused for three, four or more numbers at once, which is why it is so efficient under exam pressure.
The division method for HCF is fastest for large numbers where listing factors is painful, for instance HCF(391, 425). It needs no prime factorisation at all, so you avoid hunting for awkward prime factors of unfriendly numbers.
The product rule and fraction formulas
For any two numbers, the relationship below always holds and is a favourite shortcut.
HCF × LCM = Product of the two numbers
This is true only for two numbers, not for three or more. If you know any three of the four quantities, you can find the fourth.
Two handy fraction results follow directly:
So for the fractions 2/3 and 4/9: HCF = HCF(2,4) / LCM(3,9) = 2/9, and LCM = LCM(2,4) / HCF(3,9) = 4/3. Always reduce each fraction to its lowest terms first, otherwise the numerators and denominators you feed into the formula will give a wrong answer.
One more consequence of the product rule is worth memorising: if you double both numbers, both the HCF and the LCM also double, because every factor scales together. This proportional behaviour occasionally appears as a tricky one-liner in the paper, and knowing it lets you answer without recomputing anything from scratch.
Worked example: using the product rule
The HCF of two numbers is 12 and their LCM is 144. If one of the numbers is 36, find the other.
So the other number is 48. Quick check: HCF(36, 48) = 12 and LCM(36, 48) = 144. Both match, so the answer is confirmed.
Spotting HCF vs LCM in word problems
The hardest part is deciding which tool to use. These cues come up again and again in CDS papers.
Use HCF when…
- You must cut, divide or distribute into the largest equal groups (largest tile, longest tape, maximum students per row).
- You need the greatest number that divides given numbers leaving the same or no remainder.
Use LCM when…
- Repeating events must happen together again (bells tolling, lights flashing, runners meeting at the start).
- You need the smallest number exactly divisible by several numbers.
- Different-sized objects must be combined into the smallest equal length, area or quantity (the least cloth that can be cut exactly into pieces of given sizes).
A reliable test is to imagine making the answer bigger. If a larger answer would still satisfy the condition but you want the limit, you usually want the smallest such number, which is LCM. If a larger answer would break the condition because it can no longer divide the given numbers, you want the largest divisor, which is HCF.
Reading “maximum” and rushing to LCM, or seeing “together” and computing HCF. Slow down for two seconds and ask: am I grouping into equal parts (HCF) or am I aligning repeats (LCM)? This one habit prevents most lost marks on the topic.
Remainder-type questions
Remainder questions look intimidating but reduce to one of two clean patterns. The trick is to convert the remainder information into a clean divisibility statement, after which a normal HCF or LCM finishes the job.
Same remainder r in each case: the required greatest number = HCF of (the differences between the numbers), or HCF of (each number − r). For example, the greatest number dividing 70 and 125 leaving remainders 5 and 8 is HCF(70−5, 125−8) = HCF(65, 117) = 13.
Least number that leaves remainder r when divided by several numbers: answer = (LCM of those numbers) + r. For the least number leaving remainder 3 when divided by 4, 5 and 6, take LCM(4,5,6) = 60, giving 60 + 3 = 63.
If the remainders are different but the gap between each divisor and its remainder is constant, the answer becomes (LCM − common gap).
Previous-year style question
Q. Three bells ring at intervals of 9, 12 and 15 minutes respectively. If they all ring together at 9:00 a.m., at what time will they next ring together?
Answer: The bells coincide after LCM(9, 12, 15) minutes. 9 = 32, 12 = 22×3, 15 = 3×5, so LCM = 22×32×5 = 180 minutes = 3 hours. They next ring together at 12:00 noon.
Quick revision
- HCF = largest number dividing all; LCM = smallest number divisible by all.
- Prime factorisation: HCF uses lowest powers, LCM uses highest powers.
- Division method finds HCF fast for large numbers.
- HCF × LCM = product of the two numbers (two numbers only).
- Grouping/cutting/maximum → HCF; repeating/together/minimum → LCM.
- Least number leaving remainder r = LCM + r.
Practise five mixed word problems a day for a week and these questions will become guaranteed marks in your CDS attempt.
Frequently asked questions
Can the HCF of two numbers be greater than their LCM?
No. For any two positive integers the HCF is always less than or equal to the LCM, because HCF cannot exceed the smaller number while LCM cannot be smaller than the larger number. They are equal only when both numbers are identical.
Does the product rule HCF × LCM = product work for three numbers?
No, it holds only for exactly two numbers. For three or more numbers there is no such simple identity, so compute HCF and LCM separately using prime factorisation or the division method.
How do I quickly find the HCF of two large numbers in the exam?
Use the division (Euclidean) method: divide the larger by the smaller, then keep dividing the previous divisor by the remainder until you reach zero. The last non-zero divisor is the HCF, and it avoids full prime factorisation.
When two numbers are co-prime, what are their HCF and LCM?
Co-prime numbers share no common factor other than 1, so their HCF is 1. By the product rule, their LCM equals the product of the two numbers, for example HCF(8, 15) = 1 and LCM(8, 15) = 120.
How do I decide between HCF and LCM in a word problem?
Ask whether you are splitting things into the largest equal groups (HCF) or finding when repeating events align again (LCM). Words like maximum, largest and exactly divide point to HCF; together, smallest and simultaneously point to LCM.
Related CDS / OTA Maths topics
Want a teacher to walk you through CDS / OTA Maths?
Cavalier's CDS / OTA batches break every topic into classroom sessions with daily practice, tests and doubt-clearing.