Problems on H.C.F and L.C.M are among the most important and high-scoring topics in Quantitative Aptitude. Many students find them confusing because, unlike simple arithmetic, these questions involve factors, multiples, highest powers, prime factorizations, and different calculation methods.
If you understand these three ideas:
✔ Factors
✔ Multiples
✔ Prime power comparison
Then you can solve ANY H.C.F–L.C.M question effortlessly.
This Problems on H.C.F and L.C.M guide covers all formulas, concepts, shortcuts, cases, tables, FAQs, and exam tips making it the perfect one-stop resource for competitive exams.
Quick Overview: Problems on H.C.F & L.C.M Formulas
| Concept / Situation | Value Considered | Rule / Method Used | Formula (With Meaning of Symbols) |
|---|---|---|---|
| Factor & Multiple | Exact divisibility | Division test | If a divides b, → a is a factor & b is a multiple |
| H.C.F (Prime Factor Method) | Prime factors of each number | Least power selection | H.C.F = product of lowest powers of common prime factors |
| H.C.F (Division / Euclid Method) | Remainders from repeated division | Divide until remainder = 0 | Last non-zero divisor = H.C.F |
| H.C.F of 3 or more numbers | Pairwise H.C.F | Repeated H.C.F process | H.C.F = H.C.F( H.C.F(a, b), c ) |
| L.C.M (Prime Factor Method) | Prime factorization | Highest power selection | L.C.M = product of highest powers of all primes |
| L.C.M (Common Division Method) | Common divisibility | Divide row-wise | L.C.M = product of divisors × remaining numbers |
| Relation between H.C.F & L.C.M | Two numbers | Fundamental identity | a × b = H.C.F × L.C.M |
| Co-primes | H.C.F = 1 | Divisibility test | If H.C.F(a, b) = 1 → a & b are co-primes |
| H.C.F of Fractions | Numerators & denominators | Ratio rule | H.C.F = (H.C.F of numerators) ÷ (L.C.M of denominators) |
| L.C.M of Fractions | Numerators & denominators | Reciprocal rule | L.C.M = (L.C.M of numerators) ÷ (H.C.F of denominators) |
| H.C.F / L.C.M of Decimals | Equal decimal places | Decimal shifting | Make decimals equal → remove → find → reinsert decimal |
| Comparing Fractions | L.C.M of denominators | Equivalent fraction rule | Convert all fractions using L.C.M denominator → greatest numerator wins |
Formulas for H.C.F and L.C.M
Factor–Multiple Basics (Foundation Formula)
Before using H.C.F–L.C.M formulas, understand this fundamental rule of numbers.
1. Factor of a Number
If a number a divides b exactly, then:
a is a factor of b
Example:
6 divides 42 → 6 is a factor of 42.
2. Multiple of a Number
b is a multiple of a
Example:
42 is a multiple of 6.
Why this matters?
H.C.F focuses on common factors,
L.C.M focuses on common multiples.
Understanding this distinction is necessary for applying larger formulas.
H.C.F Formulas (Highest Common Factor)
H.C.F measures the greatest exact divisor common to all numbers.
1. Prime Factorization Method (Least Power Rule)
To find H.C.F:
- Break each number into prime factors
- Choose only primes common to all numbers
- Choose the lowest power of each
- Multiply them
Formula
H.C.F = ∏(common primes)least power
Meaning of symbols
- “∏” = product
- “Common primes” = appear in all numbers
- “Least power” = minimum exponent among numbers
Why this formula works?
Because a factor of ALL numbers can only include primes that appear in EACH number, and only up to the minimum repetition.
Small Example
12 = 22×31
18 = 21×32
Common primes = 2 and 3
Least powers = 21,31
H.C.F = 21×31= 6
Common mistakes
- Using highest powers instead of least
- Including primes that are not common
- Missing a prime factor
2. Euclid Division Method (Remainder Method)
This is the fastest method for large numbers.
Steps
- Divide larger number by smaller
- Replace larger with divisor, smaller with remainder
- Continue until remainder = 0
- Last divisor = H.C.F
Formula Logic
If:
a = bq+r
Then:
H.C.F(a,b) = H.C.F(b,r)
Example
Find H.C.F of 84 and 30:
84 ÷ 30 → remainder 24
30 ÷ 24 → remainder 6
24 ÷ 6 → remainder 0
So H.C.F = 6
Common mistakes
- Stopping too early
- Not continuing until remainder becomes 0
3. H.C.F of Three or More Numbers
Use repeated H.C.F method:
H.C.F(a,b,c) = H.C.F(H.C.F(a,b),c)
Example
Find H.C.F of 18, 24, 30:
Step 1: H.C.F(18, 24) = 6
Step 2: H.C.F(6, 30) = 6
Answer = 6
L.C.M Formulas (Least Common Multiple)
L.C.M measures the smallest number divisible by each of the given numbers.
1. Prime Factorization Method (Highest Power Rule)
Formula
L.C.M = ∏(all primes)highest power
Meaning of symbols
- “All primes” = primes appearing in any number
- “Highest power” = maximum exponent across numbers
Why this formula works?
To be divisible by all numbers, the L.C.M must include every prime, and include it to the highest extent needed.
Example
12 = 22×31
18 = 21×32
Highest powers = 22, 32
L.C.M = 22× 32 = 36
Common mistakes
- Using least powers
- Forgetting to include primes that appear only once
2. L.C.M by Common Division Method
Steps
- Write numbers in a row
- Divide by a number that divides at least two of them
- Carry forward undivisible numbers
- Multiply all divisors and remaining numbers
Formula
L.C.M = Product of divisors × undivided numbers
Example
Find L.C.M of 12, 18, 24:
Divide by 2 → 6, 9, 12
Divide by 3 → 2, 3, 4
Divide by 2 → 1, 3, 2
Divide by 3 → 1, 1, 2
Multiply: 2 × 3 × 2 × 3 × 2 = 72
Relation Between H.C.F and L.C.M
For two numbers only:
a×b = H.C.F × L.C.M
Why this works?
Both numbers share a part (H.C.F) and differ in the rest (L.C.M).
Example
If a = 20, b = 12:
H.C.F = 4
L.C.M = 60
Check:
20 × 12 = 240
4 × 60 = 240 ✔
H.C.F & L.C.M of Fractions
1. H.C.F of Fractions
H.C.F = H.C.F of numerators/L.C.M of denominators
Example
H.C.F of: 4/9, 8/27
H.C.F of numerators = H.C.F(4, 8) = 4
L.C.M of denominators = L.C.M(9, 27) = 27
H.C.F=4/27
2. L.C.M of Fractions
L.C.M = L.C.M of numerators/H.C.F of denominators
Example
L.C.M of 4/9, 8/27:
L.C.M of numerators = 8
H.C.F of denominators = 9
L.C.M=8/9
Decimals: H.C.F & L.C.M
Steps
- Equalize decimal places
- Remove decimal
- Find H.C.F or L.C.M
- Restore decimal place
Example
Find L.C.M of 0.4, 1.2
Equalize → 0.4 = 0.40
Remove decimal → 40, 120
L.C.M(40, 120) = 120
Restore → 1.20
Answer = 1.2
Comparison of Fractions
Formula Approach
- Find L.C.M of denominators
- Convert all fractions using L.C.M as new denominator
- Compare numerators
Example
Compare 3/8 and 5/12:
L.C.M(8, 12) = 24
3/8 = 9/24
5/12 = 10/24
10 > 9 →
5/12 is greater.
Smart Tips and Practical Tricks for Solving Problems on H.C.F and L.C.M
Mastering Problems on H.C.F and L.C.M becomes simple when you understand how factors, multiples, and prime powers work together. Most students make mistakes not because formulas are difficult, but because they apply them without understanding the structure of the numbers. This section breaks down the most important concepts into clear, actionable tips so you can solve questions faster and more accurately.
1. Always Convert Numbers Into Prime Factors First
Most H.C.F and L.C.M errors occur because students try to calculate directly without breaking numbers into their prime components.
Prime factorization helps you see:
- Common factors → needed for H.C.F
- Maximum powers → needed for L.C.M
This simple step improves accuracy in almost every question.
2. Use “Least Power for H.C.F, Highest Power for L.C.M” Rule
This rule instantly solves 90% of exam questions:
- H.C.F = Common primes with lowest power
- L.C.M = All primes with highest power
Always check:
✔ Which prime factors appear?
✔ What is the least power?
✔ What is the highest power?
This rule keeps your calculations clean and error-free.
3. Apply Division Method for Large Numbers
When numbers are large, factorization takes time.
Use Euclid’s Division Method:
- Divide larger number by smaller
- Continue dividing remainder into previous divisor
- Last divisor = H.C.F
This is the fastest method for big values and frequently used in exams like SSC and Banking.
4. Understand the Relationship Between H.C.F and L.C.M
For two numbers:
Product = H.C.F × L.C.M
This single formula helps solve reverse-type questions quickly.
Example: If product = 360 and H.C.F = 12
L.C.M = 360 ÷ 12 = 30
This saves time and avoids long calculations.
5. Treat Fractions Carefully With Special Formulas
Fraction questions confuse many students.
Remember:
- H.C.F of fractions = H.C.F of numerators ÷ L.C.M of denominators
- L.C.M of fractions = L.C.M of numerators ÷ H.C.F of denominators
Do NOT mix numerator and denominator roles.
6. Equalize Decimal Places Before Finding H.C.F or L.C.M
Decimal numbers must be standardized before processing.
Steps:
- Make decimal places equal
- Remove decimals
- Find H.C.F or L.C.M normally
- Reinsert decimal
Many mistakes happen when students skip step 1.
7. Understand Co-Primes Don’t Need to Be Prime Numbers
This is a frequent misconception.
Example: 8 and 15 are co-prime because:
H.C.F = 1
This trick helps solve reasoning-based questions quickly.
8. Use L.C.M to Compare Fractions Faster
When comparing fractions:
- Find L.C.M of denominators
- Convert all fractions to equivalent values
- Compare numerators
This method is quicker and avoids unnecessary decimal conversion.
9. Draw a Factor Table or Prime Tree for Clarity
Just like diagrams help in train problems, prime trees help in H.C.F–L.C.M problems.
They show:
- Shared primes
- Unique primes
- Lowest powers
- Highest powers
This reduces conceptual mistakes.
10. Practice Exam Patterns Regularly
Questions from H.C.F and L.C.M appear frequently in:
- SSC (CGL, CHSL, GD)
- Railways (RRB NTPC, Group D)
- Banking (IBPS, SBI, RBI)
- UPSC CSAT
- State-level aptitude exams
Repeated practice makes pattern identification automatic and improves speed.
FAQs About Problems on H.C.F and L.C.M
Q1. Why do we take the least powers of primes for H.C.F?
H.C.F must be a factor common to all numbers. A prime will only be common up to the smallest power appearing in every number.
Q2. Why does L.C.M use highest powers of primes?
L.C.M should contain all primes required to construct each number fully, so the highest power is taken.
Q3. Why is Euclid’s division method faster for big numbers?
It avoids prime factorization and reduces numbers step-by-step through remainders, making calculation much quicker.
Q4. Can two composite numbers be co-prime?
Yes. Co-prime means their H.C.F is 1, not that the numbers must be prime.
Q5. Why do we equalize decimal places before solving?
Decimal places indicate magnitude. Without equalizing, numbers become incomparable and lead to wrong H.C.F or L.C.M.
Q6. Why are there special formulas for fractions?
Fractions combine numerators and denominators, so separate treatment is needed to find common factors and common multiples.
Q7. Why is the H.C.F–L.C.M product formula only valid for two numbers?
Because the relationship depends on pairwise multiplication and breaks when more than two numbers are involved.
Q8. How does L.C.M help in comparing fractions easily?
Equal denominators convert the comparison to a numerator-only check, making it fast and exact.
Q9. Why is prime factorization important in H.C.F–L.C.M?
It breaks every number into its core building blocks. This makes identifying common and uncommon parts very easy.
Q10. What is the simplest way to master H.C.F and L.C.M?
Learn prime power rules, practice typical question patterns, and use the correct formula depending on the situation.