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
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 3medium
A rectangular hall has dimensions 144 m × 120 m. It needs to be tiled with square tiles of equal size such that no tile is cut. What is the side length of the largest square tile that can be used?
Practice 4medium
Two numbers are in the ratio 3:5. If their HCF is 8, what is their LCM?
Practice 5medium
The HCF of two numbers is 16 and their LCM is 480. How many pairs of such numbers exist?
Practice 6hard
The HCF of two numbers is 18. Their LCM is 1008. How many pairs of numbers satisfy this condition?
Practice 7hard
A rectangular hall has dimensions 144 m × 120 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, and how many tiles are needed?
Practice 8hard
Two numbers are in the ratio 5:7. Their LCM is 560. What is the sum of the two numbers?
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