LCM & HCF Exam Shortcuts / Tips / পৰীক্ষা টিপচ্

---

Exam Shortcuts / Tips / পৰীক্ষা টিপচ্

HCF & LCM – Memory Table

Basic definitions (মৌলিক সংজ্ঞা) : Methods to find HCF

(A) Prime Factorization Method : মৌলিক গুণনীয়ক পদ্ধতি

1. Find the prime factors of each number. প্ৰতিটো সংখ্যাক মৌলিক গুণনীয়কত বিভাজন কৰক।


2. Select the common prime factors. সকলো সংখ্যাত থকা উমৈহতীয়া মৌলিক গুণনীয়ক বাছি লওক।


3. Multiply the common prime factors to get the HCF. সেই উমৈহতীয়া গুণনীয়কবোৰ গুণ কৰিলে HCF পোৱা যায়।

Ex: Find HCF of 36 and 60

Soln

36 = 2² × 3², 60 = 2² × 3 × 5 Common factors = 2² × 3 = 12, HCF = 12

(B) Division Method (Euclid’s method)

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

Methods to find LCM

(A) Prime Factorization Method

Ex: Find LCM of 12 and 18

Soln

12 = 2² × 3, 18 = 2 × 3², LCM = 2² × 3² = 36

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

Ex:  Find LCM of 12, 18, 24

Soln: Numbers: 12, 18, 24

Divide by 2: 12 → 6, 18 → 9, 24 → 12 → New numbers: 6, 9, 12

Divide by 2 again: 6 → 3, 9 → 9, 12 → 6 → New numbers: 3, 9, 6

Divide by 3: 3 → 1, 9 → 3, 6 → 2 → New numbers: 1, 3, 2

Divide by 2: 1 → 1, 3 → 3, 2 → 1 → New numbers: 1, 3, 1

Divide by 3: 1 → 1, 3 → 1, 1 → 1 → Stop (all numbers = 1)

Multiply all divisors used: 2 × 2 × 3 × 2 × 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,

Soln: HCF = 6, so LCM = (12 × 18) / 6 = 36

HCF & LCM – Short Tricks & Memory Tips

      Three consecutive numbers: 10, 11, 12

LCM = 10×11×12 = 1320

       Three consecutive even numbers: 12, 14, 16

LCM = (12×14×16) ÷ 2 = 2688÷2 = 1344

       For fractions (a/b, c/d)

HCF & LCM of the same fractions:

Ex: HCF(2/3, 4/5)

HCF of numerators = HCF(2, 4) = 2

LCM of denominators = LCM(3, 5) = 15

HCF = 2/15

Ex: HCF(2/3, 4/5)

LCM of numerators = LCM(2, 4) = 4

HCF of denominators = HCF(3, 5) = 1

LCM = 4/1 = 4

Tip for fast memory:

Example Mixed Problem

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

Solution | সমাধান

LCM: Take highest powers= 64a³bc³

HCF: Take lowest powers= 16abc²

HCF = 16abc²  LCM = 64a³bc³

Example: Find HCF of 36 and 60

========================================

Q. Find the HCF (Highest Common Factor) of 126, 217, and 448 // Q. 126, 217 আৰু 448-ৰ মহত্তম সমাপৱৰ্তক (HCF) নিৰ্ণয় কৰক।

Options: (A) 7 (B) 14 (C) 28 (D) 56

Solution | সমাধান

Prime Factorization | মৌলিক গুণনীয়ক পদ্ধতি

• 126 = 2 × 3² × 7, 217 = 7 × 31, 448 = 2⁶ × 7

The only common prime factor is 7. (একমাত্ৰ উমৈহতীয়া মৌলিক গুণনীয়ক হৈছে 7)

HCF = 7 ​

Ans / উত্তৰ: (A) 7

Trick | কৌশল

Check divisibility by the options: 126 ÷ 7 = 18 , 217 ÷ 7 = 31 , 448 ÷ 7 = 64 

So, 14, 28, and 56 cannot be the HCF. সেয়ে 14, 28 আৰু 56 HCF হ'ব নোৱাৰে।

Ans / উত্তৰ: (A) 7

Tips | কৌশল: If one number is not divisible by an option, eliminate that option immediately. (যদি এটা সংখ্যাই কোনো বিকল্পৰে ভাগ নাযায়, তেন্তে সেই বিকল্পটো লগে লগে বাদ দিয়ক।)

================================

Q. Find the HCF (Highest Common Factor) of 105, 335, and 465. // Q. 105, 335 আৰু 465-ৰ মহত্তম সমাপৱৰ্তক (HCF) নিৰ্ণয় কৰক

Options: (A) 3 (B) 5 (C) 15 (D) 35

Trick | কৌশল

All the numbers end in 5, so first check 5 : 105 ÷ 5 = 21, 335 ÷ 5 = 67, 465 ÷ 5 = 93

Now check 15: 105 ÷ 15 = 7 : Yes,  335 ÷ 15 = 22.33 : No

So, 15 is not a common factor. Therefore, the greatest common factor is 5.

Prime Factorization | মৌলিক গুণনীয়ক পদ্ধতি : 105 = 3 × 5 × 7, 335 = 5 × 67, 465 = 3 × 5 × 31

Common prime factor = 5

HCF = 5 ​

Ans / উত্তৰ: (B) 5

Trick | কৌশল

• If all numbers end in 5, first check 5. যদি সকলো সংখ্যাৰ শেষ অংক 5 হয়, তেন্তে প্ৰথমে 5-ৰে ভাগ যায় নে নাই পৰীক্ষা কৰক।


• Test a larger common factor (15, 25, 35, etc.). তাৰ পিছত ডাঙৰ উমৈহতীয়া গুণনীয়ক (15, 25, 35 আদি) পৰীক্ষা কৰক।


• If any one number is not divisible, reject that option immediately. যদি কোনো এটা সংখ্যাই সেই সংখ্যাৰে ভাগ নাযায়, তেন্তে সেই বিকল্পটো লগে লগে বাদ দিয়ক।

Ans: (B) 5

===================================@