Learning Objectives
- Find factors and multiples of numbers
- Identify prime and composite numbers
- Understand divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, and 11
- Find HCF and LCM of numbers
- Apply prime factorisation method
Key Concepts
Factors and Multiples
A factor of a number divides it exactly without leaving a remainder. For example, factors of 12 are 1, 2, 3, 4, 6, and 12. A multiple of a number is obtained by multiplying it by a whole number. For example, multiples of 4 are 4, 8, 12, 16, 20, ... Every number is a factor and a multiple of itself. 1 is a factor of every number.
Prime and Composite Numbers
A prime number has exactly two factors: 1 and the number itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23. A composite number has more than two factors. Examples: 4, 6, 8, 9, 10, 12. The number 1 is neither prime nor composite. The number 2 is the smallest and the only even prime number.
Divisibility Rules
By 2: A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8 (even number).
By 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
By 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
By 5: A number is divisible by 5 if its last digit is 0 or 5.
By 6: A number is divisible by 6 if it is divisible by both 2 and 3.
By 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
By 10: A number is divisible by 10 if its last digit is 0.
Prime Factorisation
Expressing a number as a product of its prime factors is called prime factorisation. For example, 36 = 2 × 2 × 3 × 3. We can use factor trees or the division method to find prime factors.
HCF and LCM
HCF (Highest Common Factor): The greatest factor common to two or more numbers. Find it by listing common prime factors and multiplying them.
LCM (Lowest Common Multiple): The smallest multiple common to two or more numbers. Find it by taking the highest power of each prime factor.
Important Relationship: HCF × LCM = Product of the two numbers.
Summary
This chapter covers factors, multiples, prime and composite numbers, divisibility rules, and methods to find HCF and LCM. Prime factorisation is the foundation for finding HCF and LCM. Divisibility rules provide shortcuts to check if a number divides another exactly.
Important Terms
- Factor: A number that divides another number exactly
- Multiple: The product of a number and any whole number
- Prime Number: A number with exactly two factors (1 and itself)
- Composite Number: A number with more than two factors
- HCF: Highest Common Factor - the largest factor shared by two or more numbers
- LCM: Lowest Common Multiple - the smallest multiple shared by two or more numbers
- Co-prime Numbers: Two numbers whose HCF is 1 (e.g., 4 and 9)
Quick Revision
- 1 is a factor of every number; every number is a multiple of 1
- 2 is the only even prime number
- 1 is neither prime nor composite
- The number of factors of a number is always finite; multiples are infinite
- HCF of co-prime numbers is always 1
- LCM of two co-prime numbers = product of the two numbers
- HCF × LCM = Product of the two numbers