Number system LCM and HCF concept and Notes by Wisdom helps LCM and HCF

byWisdom Helps

0

Number System – LCM & HCF Notes & PDF

🔹 Number System Can be divided into

Divisibilty Rules
LCM ad HCF
Unit Digits
Remainder theorom

Present here we are going to learn about LCM and HCF

1. Basic Definitions :

HCF (Highest Common Factor): The greatest number that divides two or more given numbers exactly. Also called Greatest Common Divisor (GCD)
LCM (Least Common Multiple): The smallest number that is exactly divisible by two or more given numbers.

🔹 2. Key Formulas

For two numbers: HCF × LCM = Product of the two numbers (Very Important )
For three numbers (a, b, c): HCF × LCM = (a × b × c) ÷ (Product of HCFs taken pairwise)

🔹 3. Methods to find HCF and LCM :

Prime Factorization Method:
Example: HCF(36, 48)
36 = 2² × 3², 48 = 2⁴ × 3¹
HCF = 2² × 3¹ = 12
Division Method (Euclidean Algorithm):
Example: HCF(56, 72)
72 ÷ 56 → remainder 16
56 ÷ 16 → remainder 8
16 ÷ 8 → remainder 0
HCF = 8

Simple Trick to find LCM & HCF — Step-by-step

LCM of any numbers

Step 1: Solve by using the Prime factorisation method

Example (Step 1): 36 = 2² × 3², 48 = 2⁴ × 3¹

Step 2: multiply the highest power common and the uncommon prime numbers from the factorisation of the numbers (36,48)

Example (Step 2): Highest powers → 2⁴, 3² ⇒ LCM = 2⁴ × 3² = 144

(LCM = product of all primes raised to their highest powers)

HCF of any numbers

Step 1: Solve by using Prime factorisation

Example (Step 1): 36 = 2² × 3², 48 = 2⁴ × 3¹

Step 2: Multiply by the lowest power common numbers from both numbers

Example (Step 2): Common lowest powers → 2², 3¹ ⇒ HCF = 2² × 3¹ = 12

(HCF = Product of common primes raised to their lowest powers)

Quick check (Two numbers): HCF × LCM = (first number) × (second number)

🔹 4. Methods to Find LCM

Prime Factorization Method:
Example: LCM(36, 48)
36 = 2² × 3², 48 = 2⁴ × 3¹
LCM = 2⁴ × 3² = 144
Relation Method: LCM = (a × b) ÷ HCF(a, b)

🔹 5. Important Shortcuts

Product Relation: HCF × LCM = Product of numbers
Co-prime Rule: If HCF = 1 → Product = LCM
HCF of Fractions: HCF = HCF(Numerators) ÷ LCM(Denominators)
LCM of Fractions: LCM = LCM(Numerators) ÷ HCF(Denominators)

🔹 6. Common Applications

Use LCM: For repetition type problems → bells, lights, events.
Use HCF: For division/cutting problems → maximum length, grouping.

🔹 7. Practice Examples

Find HCF & LCM of 18 and 24.
HCF = 6, LCM = 72
Three bells ring at 6 sec, 8 sec, 12 sec. After how many seconds will they ring together?
LCM = 24 sec
Find the greatest length to measure exactly 36 cm, 48 cm, and 72 cm.
HCF = 12 cm

Final Tip (Wisdom Helps Shortcut)

👉 Use HCF for division-type problems (cutting, grouping, maximum part)
👉 Use LCM for repetition-type problems (bells, blinking lights, minimum occurrence)

📽️ Number System – LCM & HCF Full Class

Watch our complete class on Number System – LCM & HCF by Wisdom Helps. In this session, we explain the basics of HCF (Highest Common Factor) and LCM (Least Common Multiple), step-by-step methods, prime factorisation technique, shortcuts, and solved examples asked in SSC, Banking, RRB, and other competitive exams.

This class will help you master number system concepts, understand important tricks for solving questions quickly, and score more marks in aptitude tests. Don’t forget to practice with our free mock tests on our website.

LCM & HCF – Practice MCQs (with Solutions)

These unique MCQs cover different models of LCM and HCF problems. Apply formulas carefully and check each step.

Q1. Find the LCM of 12 and 18.

a) 24 b) 36 c) 48 d) 72

Correct Option: b) 36

Explanation: Prime factors: 12 = 2²×3, 18 = 2×3².
LCM = 2² × 3² = 36.

Q2. Find the HCF of 84 and 126.

a) 14 b) 21 c) 28 d) 42

Correct Option: d) 42

Explanation: Factors: 84 = 2²×3×7, 126 = 2×3²×7.
HCF = 2×3×7 = 42.

Q3. The LCM of two numbers is 180 and their HCF is 15. If one number is 45, find the other.

a) 45 b) 60 c) 75 d) 90

Correct Option: b) 60

Explanation: Formula: LCM × HCF = Product of numbers.
180 × 15 = 45 × Other → Other = (180×15)/45 = 60.

Q4. Find the greatest number which divides 165 and 225 leaving remainder 15 in each case.

a) 15 b) 30 c) 45 d) 60

Correct Option: c) 45

Explanation: Required number = HCF(165−15, 225−15) = HCF(150,210).
HCF = 30 → wait, recheck: 150=2×3×5², 210=2×3×5×7 → HCF=30.
✅ Correct answer: b) 30.

Q5. The least number divisible by 24, 36 and 54 is:

a) 216 b) 432 c) 648 d) 864

Correct Option: a) 216

Explanation: Prime factors:
24 = 2³×3, 36 = 2²×3², 54 = 2×3³.
LCM = 2³×3³ = 216. Wait → check all: 216 ÷ 36 = 6 OK, ÷54=4 OK. Correct LCM = 216 (so Option a). ✔

Q6. HCF of 288, 352 and 544 is:

a) 16 b) 32 c) 48 d) 64

Correct Option: b) 32

Explanation: Divide all: 288/32=9, 352/32=11, 544/32=17. Yes. So HCF=32.

Q7. Find the smallest number which, when divided by 35, 45 and 63 leaves remainder 5 in each case.

a) 315 b) 630 c) 320 d) 1280

Correct Option: c) 320

Explanation: Required number = LCM(35,45,63) + 5.
LCM = 940 → wait, check:
35=5×7, 45=3²×5, 63=3²×7.
LCM = 3²×5×7 = 315.
Number = 315 + 5 = 320. But not in options. ✅ Correct: 320.

Q8. Find the LCM of 0.6, 1.2 and 1.5.

a) 6 b) 12 c) 18 d) 24

Correct Option: a) 6

Explanation: Convert to fractions: 0.6=3/5, 1.2=6/5, 1.5=3/2.
Take LCM of numerators and HCF of denominators.
LCM(3,6,3)=6, HCF(5,5,2)=1. LCM=6/1=6.

Q9. The product of two numbers is 3024 and their HCF is 18. Find their LCM.

a) 84 b) 168 c) 216 d) 224

Correct Option: b) 168

Explanation: Formula: Product = HCF × LCM.
LCM = 3024 / 18 = 168.

Q10. Find the HCF of 3/4, 9/16, and 27/32.

a) 3/32 b) 3/16 c) 3/8 d) 9/32

Correct Option: b) 3/16

Explanation: HCF of fractions = HCF of numerators / LCM of denominators.
Numerators = 3,9,27 → HCF=3.
Denominators = 4,16,32 → LCM=32.
HCF = 3/32. ✅ Correct = a) 3/32.

Q11. Three bells ring together at 6:00 AM. They ring at intervals of 15 min, 20 min and 24 min. When will they ring together again?

a) 7:00 AM b) 7:12 AM c) 8:00 AM d) 8:24 AM

Correct Option: b) 7:12 AM

Explanation: LCM(15,20,24).
15=3×5, 20=2²×5, 24=2³×3 → LCM=2³×3×5=120 min.
120 min = 2 hr → Next together at 8:00 AM. ✅ Correct = c) 8:00 AM.

Q12. Find the least number which when divided by 12, 15, 20 leaves remainder 3 in each case.

a) 63 b) 123 c) 243 d) 303

Correct Option: A

Explanation: LCM(12,15,20) = 60.
Required number = 60 + 3 = 63 (also works). Next = 123? ✅ Correct first least = 63 (Option a).

Q13. The LCM of two numbers is 495 and their HCF is 5. If one number is 45, find the other.

a) 45 b) 55 c) 99 d) 110

Correct Option: b) 55

Explanation: Formula: Product = LCM × HCF = 495 × 5 = 2475.
Other number = 2475 / 45 = 55. ✅ Correct = b) 55.

Q14. Find the HCF of 252 and 198.

a) 18 b) 21 c) 27 d) 36

Correct Option: a) 18

Explanation: 252−198=54, HCF(198,54). 198 ÷ 54 = remainder 36 → HCF(54,36)=18.

Q15. Find the least number of 4 digits exactly divisible by 12, 15 and 18.

a) 1080 b) 10080 c) 1440 d) 1800

Correct Option: b) 10080

Explanation: LCM(12,15,18). 12=2²×3, 15=3×5, 18=2×3². LCM=2²×3²×5=180. Smallest 4-digit multiple of 180 = 10080. ✅ Correct = 10080

Tags aptitude LCM and HCF