How to Find : HCF & LCM Tricks ?

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Basic Concepts / মৌলিক ধাৰণা

1. LCM (Least Common Multiple) (সৰ্বনিম্ন সাধাৰণ গুণিতক): The smallest positive number divisible by two or more numbers. সৰ্বনিম্ন ধনাত্মক সংখ্যা যি দুটা বা তাতকৈ অধিক সংখ্যাৰে সম্পূৰ্ণভাৱে বিভাজ্য।


2. HCF (Highest Common Factor / GCD) (সৰ্বাধিক সাধাৰণ গুণনীয়ক / GCD): The greatest common factor of two or more numbers. দুটা বা তাতকৈ অধিক সংখ্যাৰ সৰ্বাধিক সাধাৰণ গুণনীয়ক। 

Formula:  LCM × HCF = Product of two number

Ex : 12 and 18 

1. HCF (12,18) = 6, LCM (12,18) = 36


2. 12×18 = 216, 6×36 = 216 Correct / সঠিক

Best Tricks for LCM / LCM

Trick 1: Prime Factorisation  / মৌলিক গুণনীয়ক ভাঙনি

    Break each number into prime factors, take the highest power of each prime, multiply. প্ৰতিটো সংখ্যা মৌলিক সংখ্যাত ভাঙি লোৱা, প্ৰতিটো মৌলিক সংখ্যাৰ সৰ্বাধিক ঘাত লৈ গুণ কৰা।

Ex: LCM (24,30,40)

1. 24 = 2³×3, 30 = 2×3×5, 40 = 2³×5


2. LCM = 2³×3×5=120

Trick 2: Division Method / ভাগ পদ্ধতি: Divide numbers simultaneously until all become 1, multiply all divisors = LCM. সংখ্যাবোৰ একেলগে ভাগ কৰি যোৱা যেতিয়ালৈকে সকলো 1 নহয়, শেষত সব ভাগকৰ গুণফল = LCM।

Trick 3: Observation / Shortcut / পৰ্যবেক্ষণ:  Multiply the largest number and adjust by smaller numbers’ factors. বড় সংখ্যাটোক লৈ গুণ কৰি সৰুবোৰৰ গুণনীয়ক বাদ দিয়া।

     Ex: LCM(24,36,48) → 48×3×1 = 144

Trick 4: Same Remainder Problems / একে অবশিষ্ট সমস্যা: If a number leaves same remainder r when divided by a,b,c → Number = LCM(a,b,c) + r. সংখ্যা যেতিয়া a,b,c লৈ ভাগ দিলে একে remainder r → Number = LCM(a,b,c) + r

         Ex: 18,20,24 remainder 3 → LCM=360 → Number=360+3=363

Trick 5: Perfect Square LCM / Perfect Square ৰ বাবে LCM:  Make all prime factor powers even to make LCM a perfect square. LCM ৰ মৌলিক সংখ্যাৰ ঘাতসমূহক যোৰ কৰি দিয়া যেন square হয়।

Ex: 16,20,24 → LCM=240 → 3600 (Perfect Square)

Best Tricks for HCF / HCF  ৰ বাবে সৰ্বশ্ৰেষ্ঠ ট্ৰিক

Trick 1: Prime Factorisation / মৌলিক গুণনীয়ক ভাঙনি:  Break each number into prime factors, take lowest power of common primes, multiply. প্ৰতিটো সংখ্যাৰ মৌলিক গুণনীয়কত ভাঙি লোৱা, সাধাৰণ মৌলিক সংখ্যাৰ সর্বনিম্ন ঘাত লৈ গুণ কৰা।

        Ex: HCF(24,36)

            24 = 2³×3, 36 = 2²×3² → HCF = 2²×3 = 12

Trick 2: Euclidean Algorithm / ইউক্লিড এলগৰিদম:  Fast for large numbers: Divide large by small → remainder → repeat until remainder 0. বড় সংখ্যা ভাগৰফল সৰু সংখ্যা, পুনৰ ভাগ কৰি অবশিষ্ট 0 হলে শেষ।

Ex: HCF(252,105)

   252 ÷ 105 = 2 remainder 42, 105 ÷ 42 = 2 remainder 21,  42 ÷ 21 = 2 remainder 0 → HCF = 21

Trick 3: Observation Method / পৰ্যবেক্ষণ পদ্ধতি:  If numbers are multiples of a common number, pick the largest number dividing all. যদি সংখ্যাসমূহে একে সৰু সংখ্যাৰ গুণিতক হয়, সকলোকে ভাগ কৰিব পৰা সৰ্বাধিক সংখ্যা লওক।

Ex: HCF(36,48,60) → divisible by 12 → HCF=12

Combined Tricks (LCM & HCF) / LCM আৰু HCF মিলিত ট্ৰিক

        LCM × HCF = Product of Numbers → Always check

LCM of three numbers (a, b, c)

    First find LCM of a and b, LCM₁ = LCM(a, b)

    Now find LCM of LCM₁ and c, LCM(a, b, c) = LCM(LCM(a, b), c)

     The final result is the LCM of all three numbers

HCF of three numbers (a, b, c)

    First find HCF of a and b, HCF₁ = HCF(a, b)

    Now find HCF of HCF₁ and c, HCF(a, b, c) = HCF(HCF(a, b), c)

    The final result is the HCF of all three numbers, Prime factorisation once → reuse for both

    Ex : Numbers: 12, 18, 24 HCF & LCM = ?

Basic definitions

1.  HCF (Highest Common Factor) is the largest number that divides two or more given numbers exactly without leaving any remainder. Example, HCF of (8, 12) = 4


2.  LCM (Least Common Multiple) is the smallest number that is exactly divisible by all the given numbers. Example, LCM of (8, 12) = 24.

Methods to find HCF

(A) Prime Factorization Method

1. Write the prime factors of each number.


2. Multiply the common prime factors with lowest powers.

Ex: Find HCF of 36 and 60
       36 = 2² × 3²
       60 = 2² × 3 × 5
       Common factors = 2² × 3 = 12
       HCF = 12

(B) Division Method (Euclid’s method)

1. Divide the larger number by the smaller number.


2. Divide the divisor by the remainder.


3. Continue until remainder = 0.


4. The last divisor = HCF.

Ex: Find HCF of 60 and 36
    60 ÷ 36 → remainder 24
    36 ÷ 24 → remainder 12
    24 ÷ 12 → remainder 0
    HCF = 12

(C) Shortcut Trick (for 2 or 3 numbers)

1. If numbers are multiples, smaller = HCF. Ex: HCF of 12, 24, 36 = 12


2. If numbers are consecutive, HCF = 1. Ex: HCF of 8, 9, 10 = 1

Methods to find LCM

(A) Prime Factorization Method

1. Write the prime factors of each number.


2. Multiply all prime factors with highest powers.

Ex: Find LCM of 12 and 18
       12 = 2² × 3
       18 = 2 × 3²
        LCM = 2² × 3² = 36

(B) Division Method (for 2 or more numbers)

1. Write numbers in a row.


2. Divide by any prime that divides at least one number.


3. Continue till all become 1.


4. Multiply all divisors = LCM

Ex: Find LCM of 12, 18, 24

Write the numbers: 12, 18, 24

Divide by 2 (smallest prime): 12 ÷ 2 = 6, 18 ÷ 2 = 9, 24 ÷ 2 = 12 → New numbers: 6, 9, 12

Divide by 2 again: 6 ÷ 2 = 3, 9 ÷ 2 = 9 (not divisible), 12 ÷ 2 = 6 → 3, 9, 6

Divide by 2 again: 3 ÷ 2 = 3, 9 ÷ 2 = 9, 6 ÷ 2 = 3 → 3, 9, 3

Divide by 3: 3 ÷ 3 = 1, 9 ÷ 3 = 3, 3 ÷ 3 = 1 → 1, 3, 1

Divide by 3 again: 1 ÷ 3 = 1, 3 ÷ 3 = 1, 1 ÷ 3 = 1 → 1, 1, 1 (Stop)

Multiply all divisors used: 2 × 2 × 2 × 3 × 3 = 72

      LCM = 72

(C) Relation Between HCF and LCM

For any two numbers,

HCF × LCM = Product of numbers

Ex: If numbers are 12 and 18, HCF = 6, so LCM = (12 × 18) / 6 = 36

Short Tricks & Memory Tips

HCF & LCM Tricks

One number divides the other:

o    HCF = Smaller number, LCM = Larger number

o    Ex: (4, 8) → HCF = 4, LCM = 8

Co-prime numbers (no common factor except 1):

o    HCF = 1, LCM = Product of numbers

o    Ex: (4, 9) → HCF = 1, LCM = 36

Three consecutive numbers:

o    HCF = 1

o    Ex: (10, 11, 12) → HCF = 1

Three consecutive even numbers:

o    HCF = 2

o    Ex: (12, 14, 16) → HCF = 2

Fractions (a/b, c/d):

o    HCF = HCF of numerators ÷ LCM of denominators

o    LCM = LCM of numerators ÷ HCF of denominators

o    Ex: HCF(2/3, 4/5) = HCF(2,4)/LCM(3,5) = 2/15

Example Mixed Problem

Find HCF and LCM of 16a²bc³, 32abc, 64a³bc²

LCM: Take highest powers = 64a³bc³

HCF: Take lowest powers = 16abc²

HCF = 16abc², LCM = 64a³bc³

Ex: Find HCF of 36 and 60

• 36 = 2² × 3²


• 60 = 2² × 3 × 5
  Common = 2² × 3 = 12
  HCF = 12

Ex

LCM & HCF – Memory Points

     First write prime factors of the numbers. Look at the powers (exponents) of common prime numbers.

For HCF

     Choose the LOWEST power of each common prime factor. Do NOT take highest power. Multiply the chosen lowest powers → HCF

For LCM

Choose the HIGHEST power of each prime factor (common or not). Do NOT take lowest power. Multiply the chosen highest powers → LCM

Ex: 12 and 24

12 = 2²×3¹

24 = 2³×3¹

HCF:
Lowest power of 2 = 2²   &  3 = 3¹
HCF = 2²×3 = 12

LCM:
Highest power of 2 = 2³  & 3 = 3¹
LCM = 2³×3 = 24

Memory Trick

HCF → Lowest power, LCM → Highest power