Factor and Multiples
Synopsis
- When two or more numbers are multiplied, a product is obtained. Each number is called a factor of the product. The product is called
- Multiples of a multiple of each of the numbers.
e.g., \[\text{5 }\times \text{ 7 }\times \text{ 9}=\text{315}\]
5, 7 and 9 are factors of 315 and 315 is a multiple of each of 5, 7 and 9.
- The multiples which appear in both the lists of multiples of the given numbers are called their common multiples.
e.g.., Multiples of 
\[\therefore \] Common multiples of 2 and 4 are 4, 8, 12, ......
- A number is said to be the factor of another if it can divide the other completely. So, a factor is also called the divisor. The factors of a number are less than or equal to the number.
- 1 is a factor of every number. Every number is a factor of itself.
- A number is the greatest factor of itself and 1 is the least factor of a number.
- Every multiple of a number is greater than or equal to the number itself.
- Every number is a multiple of 1.
- Every number is a multiple of itself.
- There is no greatest multiple of a number.
Even Number:
A number which when divided by 2, leaves zero as remainder, is an even number.
e.g., 2, 4, 6, 8, 10,...
Odd Number:
A number which when divided by 2, leaves one as remainder, is an odd number.
e.g., 1, 3, 5, 7, 9,...
Prime Number:
- A number that has only 2 factors, that is, 1 and the number itself is known as a prime number, g., 2, 3, 5, 7, 11, ...
Composite Number:
- A number which has more than two factors is called a composite number.
- g., 4, 6, 8, 9, . ..
- Number 1 has only one factor. It is neither prime nor composite. It is called a unique number.
- 2 is the only even prime number and also the smallest prime number.
- All prime numbers except 2 are odd numbers.
- All even numbers except 2 are composite numbers.
Co-prime Numbers:
- 4 Two numbers which have no other common factor except 1 are called co-prime numbers. g., 3, 4; 4,7; etc.,
Twin-prime Numbers:
- Consecutive prime numbers with a difference of 2 between them are called twin primes.
e.g., 3, 5; 5, 7; 11, 13; etc.,
The factors which divide both the given numbers are called their common factors.
e.g., Factors of 27 are 
and 27.
Factors of 18 are 
and 18.
\[\therefore \] Common factors of 27 and 18 are 1, 3 and 9.
Rules of divisibility:
- A number is divisible by 2 if the digit in its ones place is any of 0,2,4,6 or 8.
- A number is divisible by 3, if the sum of its digits is divisible by 3.
- A number is divisible by 4, if the number formed by the digits in its tens and units place i.e., last 2 digits is divisible by 4.
- A number is divisible by 5, if the digit in its ones place is either 0 or 5.
- A number is divisible by 9, if the sum of its digits is divisible by 9.
- A number is divisible by 10, if the digit in its ones place is 0.
