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Chapter Analysis
Intermediate18 pages • EnglishQuick Summary
Chapter 3, 'Finding Common Ground', explores the concepts of Highest Common Factor (HCF) and Lowest Common Multiple (LCM). Using prime factorization, students learn efficient methods to calculate these values and apply them in real-world scenarios. The chapter introduces patterns and properties associated with HCF and LCM and emphasizes their significance in simplifying mathematical problems.
Key Topics
- •Prime factorization
- •HCF (Highest Common Factor)
- •LCM (Lowest Common Multiple)
- •Properties of multiplication
- •Generalization of number properties
- •Application in real-world scenarios
- •Efficient calculation methods
Learning Objectives
- ✓Understand and apply the concept of HCF and LCM in problem-solving.
- ✓Use prime factorization to simplify calculations of HCF and LCM.
- ✓Recognize patterns and properties involving number operations.
- ✓Generalize mathematical properties using algebraic expressions.
- ✓Apply mathematical concepts to real-life situations effectively.
Questions in Chapter
Find the common factors and the HCF of the following numbers: (a) 50, 60 (b) 140, 275 (c) 77, 725 (d) 370, 592 (e) 81, 243
Page 52
Make a general statement about the HCF for the following pairs of numbers: (a) Two consecutive even numbers (b) Two consecutive odd numbers (c) Two even numbers (d) Two consecutive numbers (e) Two co-prime numbers
Page 59
Find the LCM and HCF of 225 and 750.
Page 54
Additional Practice Questions
Explain why the LCM of two numbers is never greater than the product of the two numbers.
mediumAnswer: The LCM of two numbers is derived from their prime factors. Since LCM comprises the highest power of each prime in the factorization, and the product of the two numbers includes all primes possibly multiplied by a common factor, LCM will never exceed their product, as it is a component of it.
How can you use prime factorization to find both HCF and LCM of multiple numbers efficiently?
mediumAnswer: Prime factorization allows us to identify all prime factors of the numbers. HCF can be found by taking common primes with the smallest powers, while LCM can be determined by taking all primes with the highest powers present in any number. This systematic approach simplifies the process for multiple numbers.
What is the smallest number that leaves a remainder of 10 when divided by 11 and is divisible by 3, 4, 5, and 7?
hardAnswer: The smallest such number is the LCM of 3, 4, 5, and 7, plus 10. Calculating the LCM gives us 420, adding 10 gives us 430.
Describe a scenario where finding the LCM is necessary in daily life.
easyAnswer: LCM is useful for scheduling events that repeat over different cycles, such as a bus timetable coinciding with train arrivals. Finding common multiple travel times ensures coordination.
How does doubling two numbers affect their HCF and LCM?
mediumAnswer: When two numbers are doubled, both their HCF and LCM double as well. This is because doubling a number adds another factor of 2 to its prime factorization, which impacts the HCF and LCM calculations similarly.