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 ropes of lengths 48 m, 60 m, and 72 m are to be cut into equal pieces without any waste. What is the maximum length of each piece?
Practice 3easy
Find the HCF of 144 and 180.
Practice 4easy
The HCF of two numbers is 18. If the numbers are in the ratio 2:3, find the numbers.
Practice 5easy
The HCF of two numbers is 15. If the numbers are in the ratio 3:5, what are the two numbers?
Practice 6easy
Find the HCF of 144 and 180 using the Euclidean algorithm.
Practice 7easy
The product of two numbers is 2880 and their HCF is 12. Find their LCM.
Practice 8easy
The HCF of two numbers is 15. If the numbers are 60 and 75, verify the HCF and find their LCM.
Practice 9easy
Find the LCM of 24, 36, and 48.
Practice 10easy
Three bells ring at intervals of 8, 12, and 16 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?
Practice 11easy
Three bells ring at intervals of 12, 18, and 24 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?
Practice 12medium
Two numbers have an HCF of 18. Their LCM is 1080. How many such pairs of numbers are possible?
Practice 13medium
The HCF of two numbers is 16 and the sum of the two numbers is 144. If the numbers are in the ratio 3:5, find the larger number.
Practice 14medium
Two numbers have an HCF of 15 and an LCM of 180. How many pairs of such numbers exist?
Practice 15medium
The HCF of two numbers is 16 and the sum of the two numbers is 144. If one number is 48, what is the other number?
Practice 16medium
Two numbers have an HCF of 15 and an LCM of 180. How many pairs of such numbers exist?
Practice 17medium
Four ropes of lengths 24 m, 36 m, 48 m, and 60 m are to be cut into equal pieces without any waste. What is the maximum length of each piece?
Practice 18medium
Three bells ring at intervals of 8, 12, and 18 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?
Practice 19medium
A rectangular hall has dimensions 48 m × 36 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 20medium
The HCF of two numbers is 18. If the numbers are in the ratio 2:3, find the larger number.
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