Detailed Solutions: LCM and HCF
LCM and HCF are two of the most important topics in Mathematics for SSC, Banking, Railway, NDA, and other competitive exams. These concepts are widely used in fractions, word problems, time and work, and number system questions.
Here, you will get the Top 10 Important LCM and HCF Questions with Answers that are frequently asked in exams.
What is LCM and HCF in Mathematics?
LCM (Least Common Multiple) is the smallest number that is exactly divisible by two or more numbers.
HCF (Highest Common Factor) is the greatest number that divides two or more numbers completely.
Question: 1
दो संख्याओं का ल. स. 864 तथा म. स. 144 है। यदि उनमें से एक संख्या 288 है, तो दूसरी ज्ञात करें।
The LCM of two numbers is 864 and the HCF is 144. If one of the numbers is 288, find the other.
[A] 576 [B] 1296
[C] 432 [D] 144
⚡Solution:
Answer (c) : मान लिजिए दूसरी संख्या \( x \) है।
Formula: पहली संख्या × दूसरी संख्या = ल. स. × म. स.
\(\Rightarrow \) 288 × \(x\) = 864 × 144
\(\Rightarrow \) 2 × \(x\) = 864
\(\Rightarrow \) \( x \) = 432
अतः दूसरी संख्या 432 है।
Question: 2
दो संख्याओं का ल. स. 1920 तथा म. स. 16 है। यदि एक संख्या 128 है, तो दूसरी ज्ञात करें।
The LCM of two numbers is 1920 and HCF is 16. If one number is 128, find the other.
[A] 204 [B] 260
[C] 240 [D] 320
⚡Solution:
Answer (c) : मान दूसरी संख्या \( B \) है।
Formula: पहली संख्या × दूसरी संख्या = ल. स. × म. स.
\(\Rightarrow \) 128 × \(B\) = 1920 × 16
\(\Rightarrow \) 8 × \(B\) = 1920
\(\Rightarrow \) \( B \) = 240
अतः दूसरी संख्या 240 है।
Question: 3
तीन अलग-अलग ट्रैफिक लाइटें क्रमशः 48, 72 और 108 सेकंड के अंतराल पर बदलती हैं। यदि वे सभी सुबह 8:00 बजे एक साथ बदलती हैं, तो वे फिर से किस समय एक साथ बदलेंगी?
Three different traffic lights change every 48, 72, and 108 seconds respectively. If they all change simultaneously at 8:00 AM, at what time will they change together again?
[A] 16 min [B] 24 min [C] 7min 12 Sec [D] 7 min
⚡Solution:
Answer (c) : यह पता लगाने के लिए कि वे कब एक साथ आते हैं, हमें 48, 72 और 108 का LCM (ल.स.) चाहिए।
48 = 2×2×2×2×3 = \(2^4\) × 3
72 = 2×2×2×3×3 = \(2^3\) × \(3^2\)
108 = 2×2×3×3×3 = \(2^2\) × \(3^3\)
LCM = \(2^4\) × \(3^3\) = 16 × 27 = 432 sec.
Convert seconds to minutes:
432 ÷ 60 = 7 minutes and 12 seconds
Answer: वे सुबह 8:07:12 बजे फिर से एक साथ बदलेंगी।
Question: 2
दो दूध के टैंकों में क्रमशः 120 लीटर और 180 लीटर दूध है। एक ऐसे पात्र की अधिकतम क्षमता क्या है जो दोनों टैंकों के दूध को ठीक-ठीक माप सके?
Two milk tanks contain 120 liters and 180 liters of milk. What is the maximum capacity of a container that can measure the milk of both tanks in an exact number of times?
[A] 20 [B] 60
[C] 24 [D] 12
⚡Solution:
Answer (b) : हमें वह सबसे बड़ी संख्या चाहिए जो 120 और 180 दोनों को विभाजित करती हो, जो कि HCF है।
120 = 2×2×2×3×5 = \(2^3\) × 3×5
180 = 2×2×3×3×5 = \(2^2\) ×\(3^2\) ×5
HCF = \(2^2\)×3×5 = 4×3×5 = 60
Answer: अधिकतम क्षमता 60 लीटर है।
Why LCM and HCF Questions Are Important for Exams
- Frequently asked in SSC, Railway, Banking exams
- Used in fractions, time and work, ratio, and number system
- Improves problem-solving ability
- Helps in word problems
Conclusion
Mastering LCM and HCF will help you score higher in mathematics exams. Practice these important questions daily to improve accuracy and speed.