Numbers is one of the most important topics required for competitive entrance exams. It has been observed that Question paper has 20-30% question on number system.
Classification of Numbers/Integers
1. Real Number:
Real Number are classified into rational and irrational numbers.
Rational Numbers: A number that can be expressed in the p/q where p and q are integers and q is not equal to zero is called a rational number.
Examples for irrational numbers are 3/4, -2/5, etc.
Irrational Numbers: Numbers that are not rational but which can be represented by points on the number line are called irrational numbers. Examples of irrational numbers are √2, √3, 4√5, 3√9, etc.
Numbers like π, e are also irrational numbers.
All integers are rational numbers. Integers are classified into negative integers, zero and positive integers.
In problem on numbers, we vary often use the word “number” to mean an “integer.”
3. Prime Numbers:
A number other than 1 which does not have any factor apart from one and itself is called a prime number.
Examples for prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, etc.
Every prime number greater than 3 can be written in the form of (6k+1) or (6K-1) where k is an integer.
4. Composite Numbers:
Any number other than 1, which is not prime number is called a composite number. In other words, a composite number is a number that has factors other than one and itself.
Examples for composite numbers are 4, 6, 8, 9, 10, 14, 15, etc.
The number 1 is neither prime nor composite. The only prime number that is even is 2
5. Relative Primes:
Two number are said to be relative primes or co-primes if they do not have any common factor other than 1.
For example, the numbers 15 and 16 do not have any common factor and hence they are relative primes.
None of the two number may individually be prime and still they can be relative primes. Unity is a relative prime to all numbers.
If one number is divisible exactly by a second number, then the first number is said to be a multiple of the second number.
For example, 15 is a multiple of 5; 24 is a multiple of 4.
If one number divides a second number exactly, then the first number is said to be a factor of the second number.
For example, 5 is a factor of 15; 3 is a factor of 18. Factors are also called sub-multiples or divisors.
8. Even and odd numbers:
Numbers divisible by 2 are called even numbers whereas numbers that are not divisible by 2 are called odd numbers.
Examples for even numbers are 2, 4, 6, 8, 10, etc.
Some Basic Even and odd rules:
- The sum of any number of even numbers is always even.
- The sum of any number of odd numbers (i.e, sum of 3 odd numbers, sum of 5 odd numbers etc) is always odd.
- Sum of even numbers of odd numbers (i.e, sum of 2 odd numbers, sum of 4 odd numbers etc) is always even.
- The product of any number of odd number is always odd.
- The product of any number of numbers where at least one even number is even.
9. Perfect Numbers:
A number is said to be a perfect number if the sum of ALL its factors excluding itself (but including 1) is equal to the number itself.
For example, 6 is a perfect number because the factors of 6, i.e., 1, 2 and 3 add up to the number 6 itself.
Others example of perfect number are 28, 496, 8128 etc.
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