LCM (Lowest Common Multiple) and HCF (Highest Common Factor) form a fundamental arithmetic topic that appears consistently in JKTET Paper I Mathematics. This topic tests your understanding of divisibility, factors, multiples, and the relationship between numbers—skills essential for a primary-level mathematics teacher.
For the JKTET exam, you need to master three things: finding LCM and HCF using different methods, understanding the relationship between LCM and HCF, and applying these concepts to word problems involving time, distance, and measurement. Questions typically range from direct calculation to application-based problems involving real-life scenarios relevant to the J&K context (like calculating intervals for buses, dividing resources equally among students, etc.).
The pedagogy section of JKTET also expects you to know how to teach these concepts to primary students using concrete materials and local examples, making conceptual clarity doubly important.
Key Concepts
**Factors** are numbers that divide a given number exactly (without remainder). For 12, factors are 1, 2, 3, 4, 6, 12.
**Multiples** are numbers obtained by multiplying a given number by natural numbers. Multiples of 4 are 4, 8, 12, 16, 20...
**HCF (Highest Common Factor)** is the largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).
**LCM (Lowest Common Multiple)** is the smallest number that is a multiple of two or more given numbers.
**Co-prime numbers** have HCF = 1 (example: 8 and 15). Their LCM equals their product.
**The product relationship**: For any two numbers a and b, LCM × HCF = a × b. This is a frequently tested formula.
**HCF of given numbers is always less than or equal to the smallest number**; LCM is always greater than or equal to the largest number.
**Prime factorisation method** works for both LCM and HCF: HCF uses common prime factors with lowest powers; LCM uses all prime factors with highest powers.
Formulas / Key Facts
**Core Formula:** LCM(a, b) × HCF(a, b) = a × b
**Finding HCF — Three Methods:**
1. **Listing factors method**: List all factors of each number, identify common factors, pick the highest.
2. **Prime factorisation method**: Express each number as product of primes. HCF = product of common prime factors with lowest powers.
3. **Division method (Euclid's algorithm)**: Divide larger by smaller, then divisor by remainder, repeat until remainder is 0. Last divisor is HCF.
**Finding LCM — Two Methods:**
1. **Prime factorisation method**: LCM = product of all prime factors with highest powers.
2. **Division method**: Divide numbers by common primes, continue until all quotients become 1. LCM = product of all divisors.
**Special Cases:**
HCF of co-prime numbers = 1
LCM of co-prime numbers = product of the numbers
If one number is a factor of another: HCF = smaller number, LCM = larger number
Worked Examples
**Example 1: Find HCF and LCM of 18 and 24 using prime factorisation.**
Step 1: Prime factorisation
18 = 2 × 3 × 3 = 2¹ × 3²
24 = 2 × 2 × 2 × 3 = 2³ × 3¹
Step 2: Find HCF (common primes with lowest powers)
**Example 2: Find HCF of 56 and 72 using division method (Euclid's algorithm).**
Step 1: Divide 72 by 56
72 = 56 × 1 + 16 (remainder = 16)
Step 2: Divide 56 by 16
56 = 16 × 3 + 8 (remainder = 8)
Step 3: Divide 16 by 8
16 = 8 × 2 + 0 (remainder = 0)
**HCF = 8** (last non-zero divisor)
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**Example 3: Three bells ring at intervals of 4, 6, and 8 minutes. If they ring together at 9:00 AM, when will they next ring together?**
Step 1: Find LCM of 4, 6, 8
Prime factorisation:
4 = 2²
6 = 2 × 3
8 = 2³
LCM = 2³ × 3 = 24 minutes
Step 2: Add to starting time
9:00 AM + 24 minutes = **9:24 AM**
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**Example 4: The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.**
Using formula: LCM × HCF = Product of two numbers
180 × 12 = 36 × other number
2160 = 36 × other number
Other number = 2160 ÷ 36 = **60**
Common Mistakes
**Confusing LCM and HCF**: Students often mix up which one is larger. Fix: Remember "LCM is Larger, HCF is smaller" — LCM ≥ largest number, HCF ≤ smallest number.
**Forgetting to include all prime factors in LCM**: When finding LCM, students sometimes only take common factors. Fix: LCM needs ALL prime factors (common and uncommon) with highest powers.
**Applying the product formula to three or more numbers**: LCM × HCF = a × b works only for two numbers. Fix: For three numbers, find LCM and HCF separately using prime factorisation.
**Stopping division method too early**: In Euclid's algorithm, students stop before remainder becomes zero. Fix: Continue dividing until remainder is exactly 0; the divisor at that step is HCF.
**Using wrong powers in prime factorisation**: Taking highest power for HCF or lowest for LCM. Fix: HCF = lowest powers of common factors; LCM = highest powers of all factors.
Quick Reference
HCF divides both numbers; LCM is divisible by both numbers.
LCM × HCF = Product of two numbers (for exactly two numbers only).
HCF: common primes, lowest powers. LCM: all primes, highest powers.
Co-prime numbers → HCF = 1, LCM = product.
"Ring together again" problems → Find LCM.
"Divide into equal groups" problems → Find HCF.
👥 Study this together
Invite your prep group — read the same notes, then discuss doubts in this topic's shared room.