Factors and Multiples

Prime Numbers

• A counting number (other than 1) is a prime number if it has no factors other than 1 and the number itself.
• Examples:

Composite Numbers

• Counting numbers (other than 1) which are not prime are composite numbers.
• Composite numbers have more than two factors.
• Examples:

Coprime Numbers

• Two numbers are said to be coprime if they do not have any common factors other than 1.
• Example: and

Even Numbers

• A number which is a multiple of 2 is called an even number.
• Examples:

Odd Numbers

• A number which is not a multiple of 2 is called an odd number.
• Examples:

Highest Common Factor (HCF)

• The largest common factor of two or more numbers is their highest common factor (HCF).
• Example: HCF of and is .
• The highest common factor is also known as the greatest common divisor (GCD).

Lowest Common Multiple (LCM)

• The lowest common multiple (LCM) of two or more numbers is the smallest of the common multiples of the numbers.
• Example: LCM of and is .

Things to Remember

1. is neither a prime number nor a composite number.
2. Every even number (other than ) is a composite number.
3. All prime numbers are odd, except .
4. is a factor of every number.
5. Every non-zero number is a factor of itself.
6. Every non-zero number is a factor of (zero).
7. Every number is a multiple of .
8. Every non-zero number is a multiple of itself.
9. Zero is a multiple of every non-zero number.

Additional Points

1. Prime Factorization:

   • Every composite number can be expressed as a product of prime numbers. This is called prime factorization.
   • Example:

2. Properties of Even and Odd Numbers:

   • The sum of two even numbers is even. Example:
   • The sum of two odd numbers is even. Example:
   • The sum of an even number and an odd number is odd. Example:

3. Divisibility Rules:

   • A number is divisible by if its last digit is even.
   • A number is divisible by if the sum of its digits is divisible by .
   • A number is divisible by if its last digit is or .