Chapter 11: Finding Common Ground
When do two things fit together, or meet again? In this chapter we use common factors to find the largest equal pieces (HCF) and common multiples to find when repeating things line up (LCM). Prime factorisation makes both quick, and we discover patterns such as HCF × LCM = the product of the numbers.
In this chapter you will
- find common factors and the HCF
- use prime factorisation for HCF and LCM
- find common multiples and the LCM
- make general statements about HCF and LCM
Topics
- The Greatest of All NCERT P2 3.1
HCF = the greatest common factor; it answers "largest equal piece" problems.
- Least, but not Last! NCERT P2 3.2
LCM = the lowest common multiple; it answers "when do they meet again" problems.
- Patterns, Properties, and a Pretty Procedure! NCERT P2 3.3
Consecutive numbers are co-prime; the LCM of co-prime numbers is their product.
Key words
- Common factor
- a number that divides each of the given numbers
- HCF
- highest common factor (also called GCD)
- Common multiple
- a multiple of each of the given numbers
- LCM
- lowest common multiple
- Prime factorisation
- a number written as a product of primes
- Co-prime
- numbers whose HCF is 1
- Conjecture
- a claim not yet proved
Summary
- HCF: common primes, each the fewest times; it finds the largest equal piece.
- LCM: all primes, each the most times; it finds when repeating events meet.
- Dividing both numbers by common factors step by step gives the HCF and the LCM together.
- HCF × LCM = product of the two numbers; co-prime numbers have HCF 1.