Problems on H.C.F and L.C.M
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Important Formulas & Concepts
Study MaterialProblems on H.C.F and L.C.M
Problems on H.C.F (Highest Common Factor) and L.C.M (Least Common Multiple) are among the most important topics in Quantitative Aptitude. Questions from HCF and LCM are frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and various competitive examinations.
This chapter helps students understand divisibility, factors, multiples, number relationships, and arithmetic problem-solving techniques.
Why H.C.F and L.C.M are Important?
- Frequently asked in competitive exams.
- Improves logical and numerical problem-solving ability.
- Important for fractions, divisibility, and algebraic problems.
- Useful in time-based cyclic event problems.
- Foundation for advanced arithmetic topics.
Factors and Multiples
Factors
If a number exactly divides another number without leaving any remainder, then it is called a factor of that number.
Example:
Factors of 8 = {1, 2, 4, 8}
Multiples
A multiple of a number is obtained when the number is multiplied by natural numbers.
Example:
Multiples of 8 = {8, 16, 24, 32, ...}
✔ Factors are always less than or equal to the number.
✔ Multiples are always greater than or equal to the number.
Common Multiple
A common multiple of two or more numbers is a number that is exactly divisible by each of them.
Example:
Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30...
Multiples of 5 = 5, 10, 15, 20, 25, 30...
Multiples of 10 = 10, 20, 30, 40...
Therefore, common multiples are:
30, 60, 90, 120...
Highest Common Factor (H.C.F)
The Highest Common Factor (HCF) of two or more numbers is the greatest number that divides each of them exactly.
HCF is also known as:
- Greatest Common Divisor (GCD)
- Greatest Common Factor (GCF)
Example:
HCF of 12 and 18:
Factors of 12 = 1, 2, 3, 4, 6, 12
Factors of 18 = 1, 2, 3, 6, 9, 18
Highest common factor = 6
Least Common Multiple (L.C.M)
The Least Common Multiple (LCM) of two or more numbers is the smallest number that is exactly divisible by all the given numbers.
Example:
LCM of 4 and 6:
Multiples of 4 = 4, 8, 12, 16...
Multiples of 6 = 6, 12, 18...
Least common multiple = 12
Methods to Find H.C.F and L.C.M
1. Factorization Method
Express each number as the product of prime factors.
- For HCF, take common factors with smallest powers.
- For LCM, take all factors with highest powers.
| Number | Prime Factorization |
|---|---|
| 15 | 3 × 5 |
| 25 | 5² |
| 27 | 3³ |
Therefore:
HCF = 1
LCM = 3³ × 5² = 675
2. Division Method
Used mainly for finding HCF of large numbers.
Divide the larger number by the smaller number repeatedly until remainder becomes zero.
The last divisor is the HCF.
Relationship Between HCF and LCM
Important Formula
For two numbers:
Product of Numbers = HCF × LCM
Example:
If HCF = 4 and LCM = 48
Product of numbers = 4 × 48 = 192
H.C.F and L.C.M of Fractions
HCF of Fractions
HCF of fractions =
HCF of Numerators / LCM of Denominators
LCM of Fractions
LCM of fractions =
LCM of Numerators / HCF of Denominators
Example:
Find HCF of 2/3 and 4/9
HCF of numerators = HCF(2, 4) = 2
LCM of denominators = LCM(3, 9) = 9
HCF = 2/9
Important Concepts in H.C.F and L.C.M
1. Greatest Number Dividing Given Numbers Leaving Same Remainder
Required number =
HCF of differences of numbers
Example:
Find the greatest number that divides 183, 91, and 43 leaving same remainder.
Differences:
183 − 91 = 92
91 − 43 = 48
183 − 43 = 140
HCF(92, 48, 140) = 4
2. Smallest Number Leaving Same Remainder
Required number =
LCM of divisors + remainder
Example:
Find the smallest number which leaves remainder 3 when divided by 5, 6, 7, and 8.
LCM(5, 6, 7, 8) = 840
Required number = 840 + 3
= 843
Applications of L.C.M
1. Repetition Problems
Used in problems involving repeated events like:
- Bells ringing together
- Traffic signals
- Circular track problems
- Cyclic events
2. Time Interval Problems
LCM helps determine when events occur together again.
Example:
Three bells ring every 6, 8, and 12 minutes.
LCM(6, 8, 12) = 24
Therefore, they ring together every 24 minutes.
Important Formulae and Identities
| Concept | Formula |
|---|---|
| Product of Two Numbers | HCF × LCM |
| HCF of Fractions | HCF of Numerators / LCM of Denominators |
| LCM of Fractions | LCM of Numerators / HCF of Denominators |
| Greatest Number Leaving Same Remainder | HCF of Differences |
| Smallest Number Leaving Same Remainder | LCM + Remainder |
Important Exam Tips
- Learn prime factorization thoroughly.
- Memorize divisibility rules.
- Understand when to use HCF and when to use LCM.
- Practice cyclic event problems regularly.
- Remember the product relation between HCF and LCM.
- Use HCF for greatest possible divisor problems.
- Use LCM for repeating event problems.
Problems on H.C.F and L.C.M are extremely important for competitive examinations. Strong understanding of factors, multiples, divisibility, and cyclic events helps candidates solve aptitude questions quickly and accurately.