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Numbers - Classfication Of Numbers  Numbers are collection of certain symbols or figures called digits. The common number system is ...

# 03. Aptitude Concept - Numbers Theory - Types of Numbers

## Numbers - Classfication Of Numbers

Numbers are collection of certain symbols or figures called digits. The common number system is decimal number system.

Digits:- We have 10 digits they are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Zero is called insignificant digit and 1 is called a significant digit.

Place value: Each digit in a whole number has a place value, based on its position from the right as Unit’s place, Ten’s place, Hundred’s place, Thousand’s place and so on.
Ex: 49762
Place value of 2 is (2×1)      =2
Place value of 6 is (6×10)    =60
Place value of 6 is (7×100)  =700
Place value of 9 is (9×1000)  =9000
Place value of 3 is (4×10000) =40000

Face value or Intrinsic Value:  Every digit has a face value which equals the value of the digit itself, irrespective of its place in the numeral.
In the above example, the face value of 2 is 2, the face value of 7 is 7, the face value of 6 is 6, the face value of 9 is 9 and the face value of 3 is 3.

Classification of numbers:

1. Natural Numbers:  These are the numbers that are used for counting.
In other words, all positive integers are natural numbers. (Represented by N).
These are infinite and the number 1 is the least natural number.
N = {1, 2, 3, 4, 5, 6,7…}

2Whole Numbers:  The set of numbers that includes all natural numbers and the number zero are the whole numbers
W = {0, 1, 2, 3, 4, 5 , 6 ...}
Note:  a) 0 is the only whole number which is not a natural number.
b) Every natural number is a whole number.
c) The first whole number is 0

3 .Integers :  All positive , negative numbers including zero are integers.
It includes all whole numbers along with negative numbers.
i.e., {…. -3, -2, -1, 0, 1, 2, 3…} together from the set of integers
I or Z={….-3, -2, -1, 0, 1, 2, 3…}
a) Positive Integers: {1, 2, 3, 4 …} is the set of all positive integers.
b) Negative integers: {-1, -2, -3…}  is the set of all negative integers.
c) Non-positive and non-negative integers: 0 is neither positive nor negative. So {0, 1, 2, 3…} represents the set of non negative integers while {0, -1, -2, -3…} represents the set of non-positive integers.
d) Whole numbers are nothing but positive integers and zero
e) Natural numbers consist of positive integers

4. Even Numbers:  A number which is completely divisible by 2 is called the even number.    Ex: 2,4,6,8,10, ..
In other words, such numbers have 2 as a factor when they are written as the product of different numbers
For  instance 30=2 X 3 X 5
5. Odd Numbers:  A number which is not exactly divisible by 2 is called an odd number.
Ex:  1, 3, 5, 7, 9 etc
Zero  is neither even nor an odd number.
6. Real Numbers: Real numbers are classified into two types.  They are
a) Rational Numbers:  A rational number can always be represented by a fraction of the form $\frac{a}{b}$ , where p and q are integers and q is not equal to zero.
All integers and fractions are rational numbers.
Rational numbers that are not integral will have decimal values. The terminating and non-terminating periodic fractions are rational numbers. All recurring decimals are rational numbers.
Q={$\frac{a}{b}$ | p, qÎI & q≠0}
Ex:  2, 3, $\frac{3}{2}$, 4, 6, 0.666.. , $\frac{22}{7}$
b) Irrational Numbers:  An irrational number cannot be expressed in the form of p/q where p≠0.
All non terminating and non-periodic fractions are irrational numbers.
7. Imaginary Numbers: Some numbers cannot be expressed such as the square root of a negative number. An imaginary number is denoted by i. In other words, those are not real numbers
8. Complex Numbers: A combination of real numbers and imaginary numbers is known as a complex number. {5 + 2i}
9.  Factors & Multiples:
The number n which exactly divides another number m, then n is a factor of m and m is a multiple of n.
Let x and y be two integers, if x divides y completely, means the remainder is zero, then we can say that x is a factor of y and y is called multiple of x.
Example, 18 and 3, where 3 divides 18 completely, hence 3 is a factor of 18.
18 is a multiple of 3
Factors of 18 : 1,2,3,6,9,18
Multiples of 3: 3,6,9,12,15,18,…

10.  Prime numbers:
A number greater than 1 is called a prime number, if it has exactly two factors namely 1
and the number itself.
There are 15 primes numbers in first 50 natural numbers and 25 prime numbers in the first 50
Natural numbers.
2 is the only even prime number.
All prime number greater than 3 can be written in the form of 6k + 1 or 6k - 1
(where k=1,2,3,..)
Prime numbers up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

11. Composite Number: A number which has at least 3 different factors or a number having factors other than 1 and itself is a composite number.

12. Relatively Prime or Co-Prime Numbers:
If two or more numbers have no common factors other than 1 than they are said to be
relatively prime or co-primes. In other words, the HCF (Highest common factor) of the numbers
is 1.
Example, 14 and 15, the Factors of 14 are 1, 2, 7, 14 and the factors of 15 are 1, 3, 5, and 15. There are no common factors other than 1 between 14 and 15, hence they are said to be relatively prime.
Ex: (2, 3), (4, 5), (7, 9), (8, 11) etc are co-primes

13. Twin Primes: -A pair of prime numbers is said to be twin primes if they differ by 2.
Ex: -5 and 7 are twin primes.
Important points
1. The only even prime number is  2
2. 1 is neither a prime nor a composite number
3. If p is a prime number then for any whole number a, ap – a is divisible by p.
4. The remainder when a prime number p≥5 is divided by 6 is 1 or 5.
5. For prime numbers p>3, p2-1 is divisible by 24.
6A number is divisible by ab only when that number is divisible by each one of a and b, where a and b are coprime.
7. All prime numbers can be expressed in the form 6n-1 or 6n+1, but all numbers that can be expressed in this form are not prime
8. A composite number can be uniquely expressed as a product of prime factors

The process to Check A Number s Prime or not:
Take the square root of the number.
Round of the square root to the next highest integer and let us name it as  Z.
Check for divisibility of the number N by all prime numbers below Z.
If there is no numbers below the value of Z which divides N then the number will be prime.

14. Factorial:-
The factorial, symbolized by an exclamation mark (!), is a quantity defined for all integer s greater than or equal to 0.
For an integer n greater than or equal to 1, the factorial is the product of all integers less than or equal to n but greater than or equal to 1. The factorial value of 0 is defined as equal to 1. The factorial values for negative integers are not defined.
Mathematically, the formula for the factorial is as follows. If n is an integer greater than or equal to 1, then
n!n ( n - 1)( n - 2)( n - 3) ... (3)(2)(1)

Basic Formulae:
1)  (a+b)=a2+b2+2ab
2)  (a-b)=a2+b2-2ab
3)  (a+b)2-(a-b)2=4ab
4) (a+b)2+(a-b)2=2(a2+b2)
5) (a2-b2)=(a+b) (a-b)
6) (a+b+c)2=a2+b2+c2+2(ab+bc+ca)
7) (a3+b3+c3-3abc)=(a+b+c)(a2+b2+c2-ab-bc-ca)
8)  a3+b= (a+b) (a2-ab+b2)
9)  a3-b= (a-b) (a2+ab+b2)
10)   If a+b+c=0, then a3+b3+c3=3abc
11)  Sum of first n natural numbers  =  $\frac{n(n+1)}{2}$
12) Sum of the squares of first n natural numbers = $\frac{n(n+1)(2n+1)}{6}$
13) Sum of the cubes of first n natural numbers = $\frac{n^{2}(n+1)^{2}}{4}$
14) Sum of the first n even natural numbers = n(n+1)
15) Sum of the first n odd natural numbers= n

#### Some Important properties of Numbers and Divisibility

1) If z divides both x and y, then (x + y) and (x - y) are divisible by z.
Ex: 2 divides both 4 and 12, so (4 + 12) and (4 -12) will both be divisible by 2.

2) Sum of 5 consecutive whole numbers is always divisible by 5.
Ex: 1 + 2 + 3 + 4 + 5 = 15, hence divisible by 5.
3) The product of 3 consecutive natural numbers is divisible by 6.
4) The product of 3 consecutive natural numbers the first of which is even is divisible by 24.
5) Any number written in the form of 10n-1 is divisible by 3 and 9.
3) The product of three consecutive numbers, if the first number is even, the result will always be divisible by 24. Because the above numbers will always have factors 8 and 3.
Example: 2 x 3 x 4 = 24 or 4 x 5 x 6 = 120, both  the numbers, 24 and 120 are divisible by 24.
4) The product of three consecutive numbers, if the first number is odd, then the result will always be divisible by 6. Because the above numbers will always have factors 2 and 3.
Ex: 3 x 4 x 5 = 60 or 5 x 6 x 7 = 210, both the numbers, 60 and 210 is divisible by 6.
5) Difference between a number and the number formed by writing its digits in reverse order is divisible by 9.
Ex: 4321 - 1234 = 3087, which is divisible by 9.
6) Any number (10n) - 1 is divisible by 9.
Ex: 103 - 1 = 1000 - 1 = 999
7) When n is odd , n( n2 – 1 ) is divisible by 2
Ex:  n  =  9 then n(n2-1)  =  9(92 – 1)  =  720 is divisible by 24
8) If n is odd, 2n + 1 is divisible by 3, e.g. n=5,
25+1 =33, which is divisible by 3
And if n is even, 2n – 1 is divisible by 3, e.g. n=6,    26-1 =63, which is divisible by 3
9) If n is prime, then n (n4-1) is divisible by 30, e.g. n=3, 3(34-1) = 240, which is divisible by 30
10) If n is odd, 22n + 1 is divisible by 5, e.g. n=5, 22*5+1 =1025, which is divisible by 5
And if n is even, 22n – 1 is divisible by 5, e.g. n=6, 22*6-1 = 4095, which is divisible by 5
11) If n is odd, 52n + 1 is divisible by 13, e.g. n=3, 52*3+1 =15626, which is divisible by 13
And if n is even, 52n – 1 is divisible by 13, e.g. n=4, 52*4-1 =390624, which is divisible by 13
12) xn + yn  =  ( x + y ) ( xn-1 – xn-2 y +  …. + yn-1 ),
xn + yn is divisible by x + y when n is odd.
13). xn - yn  =  ( x + y ) ( xn-1 – xn-2 y +  …. - yn-1 ) when n is even,
so xn - yn is divisible by (x+y)
14). xn - yn  =  ( x - y )( xn-1 + xn-2 y +….+ yn-1 ) when n is either odd or even, so (xn - yn ) is divisible by (x-y)
15) 3n will always have an even number of tens.
16) A sum of 5 consecutive whole numbers will always be divisible by 5.
17) . The product of 3 consecutive natural numbers is divisible by 6.

18).The product of 3 consecutive natural numbers the first of which is even is divisible by 24.

Division Algorithm: -
For any two natural numbers, a and b there exists unique numbers q and r called Quotient and remainder are respectively such that  a=bq+r where 0≤r≤b.
Dividend=(Divisor  x  quotient) +Remainder