Number Properties  Atreya Roy

Author: Atreya Roy is pursuing his BTech From Kalyani Government Engineering College, Bengal.
Numbers can be categorized into different types such as composite, prime, natural, whole, integers, fractions, rational , irrational and more.
Whole Numbers : 0,1,2,3,4….
Natural Number : 1,2,3,4…….
Prime number : Number which have only 2 factors (1 and the number itself) Eg : 2,3,5,7…
All prime numbers are odd except 2. Since otherwise, they would be a multiple of 2 and will end up having more than 2 factors.
Factors : A number that completely divided a greater or equal number. Example : 2 is a factor of 16, 5 is a factor of 15etc.
Any number that can be written in p/q form can be said to be a fraction.
Concept : Properties of 0 :
1) Zero is neither negative nor positive.
2) For any Number X, apart from 0, X^{0 }= 1
3) 0^{0 }= Undefined
4) Any number when divided by 0 is undefined. X / 0 = Undefined
Concept : Prime Numbers :
1) 2 is the first prime number
2) All primes except 2 are odd
3) Any prime number greater than 3 can be expressed as : 6k+1 or 6k1, but the reverse is not always true.
4) Given below is the number of primes for different ranges :
Number Range Number of Primes
1100 25
101200 21
201 300 16
How To find out that a Number is Prime or not.
Take the root of the number and look for the number of primes below it. Check the divisibility with them. If divisible by any one, Number is not prime, otherwise, the number is prime.
Example : Is 101 a prime number ?
Solution :
Root 101 = 10.xx
Primes below 10 = 2,3,5,7
101 is not divisible by any of these. Hence 101 is a prime number.
Summation of Numbers :
 The sum of first n consecutive natural numbers 1+2+3+4….n = n(n+1)/2
 Sum of the first n consecutive squares = n(n+1)(2n+1)/6
 Sum of the first n consecutive cubes = ((n(n+1)/2)^{2}
Concept : Perfect Squares
Let us consider the first few single digit perfect squares and their original number.
1^2 =1
2^2 =4
3^2 = 9
4^2= 16
5^2 = 25
6^2= 36
7^2=49
8^2 = 64
9^2= 81
10^2 = 100
So from the above observation, it can be inferred that no perfect squares end with : 2,3,7,8
So if the unit digit of a perfect square is any of 2,3,7 or 8 , the number is not a perfect square.
More Properties of Perfect Squares :
 If the unit digit of a perfect square is 1, the “tens” digit must be even. Example : 1^2 = 01, 9^2 = 81, 11^2 = 121
 If the unit digit of the perfect square is 5, then the tens digit must be 2.
 If the Unit digit of a perfect square is 6, then the tens digit must be “odd”
 The number of zeroes present at the end of any perfect square cannot be “odd”. They must be in the form of 2k, where k =0,1,2,3…..
 A perfect Square of the form “aabb” is 7744 ( 88 ^{2} = 7744 )
Concept : Coprimes and Application of Euler
In CAT, GMAT, IBPS, SSC and other similar examinations we will face questions where will be given a range and asked to find out the multiples of certain primes or their multiples. It will be time consuming if we apply the fundamentals of progression here. So today we will learn a new concept of how to solve these sums quickly.
CoPrime : Coprime numbers are those which do not contain any other common factor other than 1 between them . Example : 2 and 3 are co prime, but 3 and 12 are not.
We will use the concept of Euler here, since we are having co – primes.
Euler Number of any prime which will be used in these sums will be viewed as = N *(1 (1/a))*(1 (1/b)) , where a,b are the primes present in the number . N is the original number. So Euler of 2 = (1(1/2)) = 1/2
To find the Euler number of greater numbers which aren’t prime, we will factorize them and then consider the primes which are in them.
Example : Euler number of 12 = ?
Solution : 12= 2^2 * 3 , So the primes present are : 2 and 3. Hence Euler Number of 12 = 12 * (1 (1/2)) * (1(1/3)) = 12 *(1/2)*(2/3) = 4
Let us look at an example that will help us understanding the application.
Example :
Find the number of numbers which are co prime to 120 and less than it ?
Solution:
2 and 3 are coprimes. Hence Total such numbers= 120 * (1 (1/2)) * (1(1/3)) = 40
Sum of coprimes :
Example : Find the sum of all co primes to 144 less than it.
Solution :
Let E signify the euler number
E(144) = 144 * 1/2 * 2/3 = 48
Sum of all co primes = E(N) * N /2 = 48 * 144/2 = 48 *72
= 3456
Product of coprimes :
The number of ways of writing a number N as a product of two coprime numbers = 2^(n1) where n=the number of prime factors of the number.
Example :
In how many ways can 30 be written as product of 2 co primes ?
Solution :
Factors of 30 = 2*3*5 = 8 factors
Prime factors = 2,3,5 = 3
Ways = 2^(31) = 2^2 = 4 ways
Concept : Writing a number as the product of 2 natural numbers
In how many ways can we write a number as a product of 2 natural numbers ? Let us take an example to learn this idea
Example:
In how many ways can we write 80 as a product of 2 natural numbers ?
Solution :
Factors of 80 = 8*10 = 2^3 * 2*5 = 2^4 * 5
Total factors = (4+1) * (1+1) = 5*2 = 10 factors
80 can be written as :
1*80
2*40
4*20
5*16
8*10
Hence 5 ways.
Proceeding with the number of factors we can say that ,
total ways that a number can be expressed as product of 2 natural numbers = N/2 where N is the number of factors in the given number. Here N is even.
In case of the total number of factors in the number is odd, then number of ways = (N+1) /2
In case the question comes that in how many ways can we write a number as the product of 2 integers, here we can also use negative numbers hence the ways will be doubled. Since a * b can also be written as : (a) * (b)
Concept : Writing a number as difference of Squares
We will often face questions where we need to find the ways in which we can write a number as difference of squares of natural numbers.
Example : In how many ways can 45 be written as the difference of squares of 2 natural numbers ?
Solution :
45 = 1*45
= 3*15
= 5*9
We can write it as :
45 = x^2 – y^2
So, (x+y)*(xy) = 45
Taking into consideration the above divisors,
x+y = 45, xy = 1 : x= 23, y=22 is a pair.
x+y = 15, xy = 3 : x = 9, y = 6 is a pair
x+y = 9 , xy = 5 : x=7 , y=2 is a pair
Hence there are 3 ways .
If the number is odd, then calculate the factors of the number.
Let the number of factors be = N
Total number of ways = N/2
Example :
In how many ways can 945 written as the difference of squares of 2 natural numbers ?
Solution :
945 = 3^3*5*7
Factors = 4*2*2 = 16 factors
Number of Ways = 16/2 = 8 ways
What if the number is even ?
We will always consider the even factors, if the number is even.
In the number is even, we will follow the following rule :
Since we will always consider even number , let the factors be 2a and 2b
So , 2a*2b = N ( N is out number )
4ab = N
Ab = N/4
Let N/4 = M
Now calculate in how many ways can M be written as the product of 2 natural numbers.