Number System: Properties of Integers
Here we are going to learn about Digits, Integers, Even and Odd Integers, Operation on even and Odd integers, Factors & Multiples, Prime Numbers & Composite Numbers.
In the next article on properties of integers, we will learn about how we use certain properties of integers to solve math problems :Commutative property of addition, Commutative property of multiplication, Associative property of addition, Associative property of multiplication, Distributive Property, Identity Property for Addition, Identity Property for Multiplication, Inverse Property for Addition and Zero Property for Multiplication.
Digits : A single symbol (such as “1” or “6”) used alone, or in combinations (such as “16”), to represent numbers/numerals (such as the number 16) according to numeral systems is called a Digit.
We use 10 digits in our day to day lives : 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
For e.g.: The numeral 1284 is made up of the digits “1”, “2”, “8” and “4”.
The numeral 6 is made up of only one digit “6”.
In the number system, position of digits is very important.
For e.g. : The number 1284 can be represented in a place value table as below.
Place Value table representing number 1284
| Thousands | Hundreds | Tens | Ones |
| 1 | 2 | 8 | 4 |
Consecutive Integers : Integers that follow a sequence, where difference between two successive integers is 1.
It can be expressed in the form : n, n+1, n+2, n+3, …….. where ‘n’ is an integer.
For e.g.: 13,14,15,16,17,…….. are consecutive integers.
Even Integers/Numbers :
An even integer is any integer that has no remainder when divided by 2. When divided by 2 such numbers results in an integer.
It can be expressed in the form ‘2n’ where ‘n’ is an integer.
Also, Consecutive even integers can be expressed in the form 2n, 2n+2, 2n+4, 2n+6,….. where ‘n’ is an integer.
For e.g. : -16, -4, 0, 2, 10 are even integers.
-2, 0, 2, 4, 6 are consecutive even integers.
Note :
~ 0′ is a even integer.
~ If an even number is divided by two, the result is another integer.
~ An even integer always ends in 0, 2, 4, 6, or 8.
Odd Integers/Numbers :
An odd integer is an integer when divided by two, either leaves a remainder or the result is a fraction.
It can be expressed in the form ‘2n+1’ where ‘n’ is an integer.
Also, Consecutive odd integers can be expressed in the form 2n+1, 2n+3, 2n+5, 2n+7,….. where ‘n’ is an integer.
For e.g. : -15, -3, 1, 3, 11 are odd integers.
-1, 1, 3, 5, 7 are consecutive odd integers.
Note:
~ An integer that is not an odd number is an even number.
~ An odd number, when divided by two, will result in a fraction.
~ An odd integer always ends in 1, 3, 5, 7, or 9.
How to identify if an integer is odd or even just by looking at the digit in one’s place?
An even integer always ends in 0, 2, 4, 6, or 8.
An odd integer always ends in 1, 3, 5, 7, or 9.
To check whether an integer is even or odd, look at the digit at the ones place. If it ends in an even integer, the given integer is also even. If it ends in an odd integer, the given integer is also odd.
For e.g. : Integer 1248 is an even integer because it ends in ‘8’, an even integer.
Similarly, Integer 1249 is an odd integer because it ends in ‘9’, an odd integer.
Operations on even and odd integers: It is elaborated in the following table.
| Addition | Example |
| Even + Even = Even | 2 + 4 = 6 |
| Odd + Odd = Even | 3 + 7 = 10 |
| Even + Odd = Odd + Even = Odd | 2+7 = 9 |
| Multiplication | Example |
| Even x Even = Even | 2 x 4 = 8 |
| Odd x Odd = Odd | 3 x 7 = 21 |
| Even x Odd = Odd x Even = Even | 2 x 7 = 14 |
Factors and Multiples :
An integer ‘x is called a factor or a divisor of another integer ‘y’ if ‘x’ divides ‘y’ completely and remainder is ‘0’. Also ‘y’ in this case is called a multiple of ‘x’.
For e.g. : 1, 2, 3, 6, 9 and 18 are factors/divisors of 18.
18, 36, 54, 72, ….. are multiples of 18.
Prime Numbers :
A prime number is a positive integer that has exactly two factors, 1 and itself.
For e.g. :17 has exactly two factors, 1 and 17. So 17 is a prime number.
18 has 6 factors 1, 2, 3, 6, 9 and 18. So 18 is not a prime number.
Note:
~ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 … are prime numbers.
~ 2 is the only even prime number.
~ Any prime number greater than 3 can be expressed as 6k+1 or 6k-1, where k is a positive integer.
~ All the numbers of the form 6k+1 or 6k-1, where k is a positive integer, are not necessarily prime numbers.
For e.g. :
Prime number, 7 = 6×1 + 1
Prime number, 13 = 6×2 + 1
Prime number, 17 = 6×3 – 1
But, 35 = 6×6 -1 is not a prime number
Composite Numbers :
Any number greater that has more than two factors i.e. other than 1 and itself is known as composite number.
For e.g. 6, 12, 18, 25, 96 …. are composite numbers.
Note:
~ So, by above definitions of prime and composite numbers, ‘1’ is neither a prime number nor a composite number.
