Core ConceptRead this first — the foundation of the topic
LCM (Least Common Multiple) and HCF (Highest Common Factor) are fundamental concepts in Number System that appear in almost every SSC CGL paper. Understanding these concepts is crucial for solving time and work, ratio, and simplification problems. Core Concept:
HCF is the largest number that divides all given numbers completely. LCM is the smallest number that is divisible by all given numbers. Think of HCF as the biggest common piece, and LCM as the smallest common whole.
Key RulesCore rules you must know cold
1
For two numbers a and b: HCF × LCM = a × b
2
HCF is always less than or equal to the smallest number
3
LCM is always greater than or equal to the largest number
4
If two numbers are co-prime (HCF = 1), then LCM = product of numbers
Formula BlockMemorise — at least one formula appears in every paper
HCF × LCM = Product of two numbers
For three numbers: HCF(a,b,c) × LCM(a,b,c) ≠ a × b × c
HCF by Division Method: Keep dividing larger by smaller until remainder is zero
LCM = (a × b) ÷ HCF(a,b)
Exam PatternsWhat examiners ask — read before attempting PYQs
SSC CGL typically asks
direct HCF/LCM calculation, word problems involving bells/traffic lights, finding numbers when HCF/LCM is given, and ratio-based problems. Most questions are 2-mark difficulty level
Powerful Shortcut - Product Rule
For any two numbers, if you know any three values among {number1, number2, HCF, LCM}, you can find the fourth instantly using: HCF × LCM = number1 × number2.
Worked ExampleSolve this step-by-step before moving on
LCM = product of highest powers = 2⁴ × 3² = 16 × 9 = 144
Verification: HCF × LCM = 24 × 144 = 3456 = 48 × 72 ✓
Worked Example 2:
Two bells ring at intervals of 15 and 20 minutes. If they ring together at 9:00 AM, when will they ring together next?
1
Step 1
Find LCM of 15 and 20
15 = 3 × 5
20 = 2² × 5
2
Step 2
LCM = 2² × 3 × 5 = 60 minutes
3
Step 3
Next time = 9:00 AM + 60 minutes = 10:00 AM
ShortcutsUse these to save 30–60 seconds per question
Another Speed Trick - Co-prime Check
If two numbers have no common factors except 1, their HCF = 1 and LCM = their product
Examples
(7,11), (15,16), (25,28).
Exam TrapsCommon mistakes students make — avoid these
Students confuse the formula application for three numbers. The golden rule HCF × LCM = product applies ONLY to two numbers. For three or more numbers, this formula does NOT work.
Always work with pairs or use prime factorization method for multiple numbers. This mistake costs students 2-4 marks per exam.
Practical Tip:
In word problems, LCM gives 'together again' scenarios (bells, traffic lights). HCF gives 'maximum equal groups' scenarios (distribution problems).
Identify the scenario type first, then apply the appropriate concept.
Key Points to Remember
HCF × LCM = Product of two numbers (works only for exactly two numbers)
HCF is the largest common divisor, LCM is the smallest common multiple
For co-prime numbers: HCF = 1, LCM = product of the numbers
LCM = (a × b) ÷ HCF for any two numbers a and b
HCF ≤ smallest number, LCM ≥ largest number among given numbers
The HCF of two numbers is 18. If the numbers are 54 and 90, verify this is correct. What is their LCM?
Practice 2easy
Three numbers are in the ratio 2:3:4. Their HCF is 5. Find the largest number.
Practice 3easy
The HCF of two numbers is 18. If the numbers are 54 and 90, verify which statement is correct.
Practice 4easy
What is the LCM of 24, 36, and 48?
Practice 5easy
Find the LCM of 24, 36, and 48.
Practice 6medium
Three bells ring at intervals of 18 minutes, 24 minutes, and 30 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?
Practice 7medium
Two numbers have an HCF of 18 and an LCM of 540. How many pairs of such numbers are possible?
Practice 8medium
The HCF of two numbers is 16. If the numbers are in the ratio 3:5, what is their LCM?
Practice 9medium
Three bells ring at intervals of 18 minutes, 24 minutes, and 36 minutes respectively. If they ring together at 10:00 AM, at what time will they ring together again?
Practice 10medium
A rectangular hall has dimensions 48 m × 36 m. It is to be paved with square tiles of the largest possible size such that no tile is cut. What is the side length of each tile?
Practice 11medium
Two numbers have an HCF of 18. Their LCM is 216. How many pairs of numbers satisfy these conditions?
Practice 12medium
Three numbers are in the ratio 2:3:4. Their LCM is 144. Find the sum of the three numbers.
Practice 13hard
The LCM of two numbers is 180 and their HCF is 6. If the sum of the two numbers is 66, find the difference between them.
Practice 14hard
Four bells ring at intervals of 6, 8, 12, and 18 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?
Practice 15hard
Two numbers have an HCF of 15 and an LCM of 1260. How many such pairs of numbers exist?
Practice 16hard
The LCM of two numbers is 12 times their HCF. If the sum of the LCM and HCF is 156, find the HCF.
Practice 17hard
Three numbers are in the ratio 2:3:4. The HCF of these three numbers is 8. What is their LCM?
Practice 18hard
Three bells ring at intervals of 18, 24, and 30 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again for the second time?
60-Second Revision — LCM and HCF
Formula: HCF × LCM = a × b (only for two numbers, not three)
Remember: Prime factorization uses lowest powers for HCF, highest for LCM
Trap: Three number product rule does NOT apply
Shortcut: Co-prime numbers have HCF = 1, LCM = their product