Numbers concept

Major concepts in numbers that we come across in various competitive exams are

Face value :  face value of a digit in a number is its own value at whatever place it may be either in units digit or tens digit or hundreds digit etc.

Example:    let us consider a number 7852

Here the face value of 7 is 7

                 face value of 8 is 8

                  face value of 5 is 5

                  face value of 2 is 2

Place value:  place value is also called as local value.

Place value varies according to the place of the digit.

Place value of unit digit in a number is = (unit digit * 1)

Place value of tens digit in a number is = (tens digit * 10)

Example: let us consider a number 7852

Here the place value of 7 is (7*1000=7000)

                 place value of 8 is (8*100=800)

                  place value of 5 is (5*10=50)

                  place value of 2 is (2*1=2)

Natural numbers:  1, 2, 3…………..

Whole numbers:   0, 1, 2, 3………..

Integers:                 ….-3, -2, -1, 0, 1, 2, 3….

Even numbers:      2, 4, 6, 8…….

Rule to check whether a number is even or not is simply divide the number with 2.

Formula for sum of even numbers is n(n+1).

Don’t know how memorize this formula first let us see how this came.

For example let us consider sum of first 3 even numbers

                       2+4+6=12

 Here 3 numbers are added right so n=3

12=3*4=3*(3+1)

             =n*(n+1)

Odd numbers:   1, 3, 5, 7……

Odd numbers are not divisible by 2

Formula for sum of odd numbers is (n^2)

Apply the same method which I stated above 

Prime numbers: A number which is divisible by 1 and itself is a prime number.

Example: 2, 3, 5, 7, 11, 13…..

You may find easy to guess whether it is prime or not if it is below 100

I think you may be wondering how to know a number greater than 100 is prime or not.

I will give you a simple logic J

For example if you want to know whether 137 is prime or not

I think it’s easy to guess the perfect square root of a number.

So here 137 lies between 121 and 169

(12)²< 137 < (13)²

So now the prime numbers below 12 are 2, 3, 5, 7, 11

Now divide 137 with these numbers if it is divisible with atleast any one of them then it is not a prime number.

ITS SIMPLE RIGHT JJ

Co primes: two numbers are said to be co-prime if their HCF is 1

Example (2, 3)

                (4, 5)

                (7, 9)

Divisibility rules:

Divisibility by 2: A number is divisible by 2 if it’s unit digit                     contains 0, 2, 4, 6, 8

Divisibility by 3: A number is divisible by 3 if sum of all its digits is      divisible by 3        

Example:    7852:7+8+5+2=22

                     22 is not divisible by 3 so 7852 is not divisible by 3

Divisibility by 4: A number is divisible by 4 if last 2 digits are      divisible by 4

Example:   7852:  here last 2 digits 52 is divisible by 4 so 7852 is 

                    divisible by 4

Divisibility by 5: A number is divisible by 5 if it’s last digit is either 0 or 5

Divisibility by 6: A number is divisible by 5 if both divisibility rules of 2 and 3 are satisfied

Divisibility by 8: A number is divisible by 8 if last 3 digits are      divisible by 8

Example:   7852:  here last 3 digits 852 is not divisible by 8 so 

                    7852 is not divisible by 8

Divisibility by 9: A number is divisible by 9 if sum of all its digits is      divisible by 9         

Example:    7852:7+8+5+2=22

                     22 is not divisible by 9 so 7852 is not divisible by 9

Divisibility by 11: A number is divisible by 11 if the difference between sum of even digits and odd digits is divisible by 11

Example 7852: sum of even digits is 7+5=12

                           Sum of odd digits is 8+2=10

           Difference of both is 12-10 is 2

So it is not divisible by 11.

Power cycle:

Number	Repeating pattern
0	0
1	1
2	2, 4, 6, 8
3	3, 9, 7, 1
4	4, 6
5	5
6	6
7	7, 9, 3, 1
8	8, 4, 2, 6
9	9, 1

Confused with this and want to know about it then let’s see this

In most of the latest placement papers we find questions like

$ Find the last digit of (135647)^34.

Most of you may think its difficult to solve so now lets solve it using this power cycle concept in few seconds.

From the above table of power cycle

The  power cycle of 7 repeats after every 4 terms i.e, 7,9,3,1

   7^1=7

   7^2=49

   7^3=343

   7^ 4=2401

   7^ 5=16807

   7^ 6=117649

.. and so on…

 Did you find the last digits of above numbers are repeating 7,9,3,1….

So the power cycle of 7 is 4

 Now for the above example the power is 34

The remainder when 34 divided by 4 is 2 .from power cycle the 2^nd term is 9 there the answer is 9.

     The last digit of (135647)^34 = 9

                           

Determine the remainder when 24^81 is divided by 23?      (^ means power)

Solution with Number System - Divisibility Shortcut trick:

To solve the question, first divide 24 by 23. The remainder is 1.

Now you have (1)^81 which equals 1. So the remainder is 1. Ans.

Important Note of Number System - Divisibility Shortcut trick: You have to convert Numerator to 1 or -1 value by dividing it with denominator. We have applied this formula in the above and below questions.

Example Question 2:

Determine the remainder when 2^24 is divided by 7?

Easy Solution of this Number system problem with Divisibility Shortcut trick:

Given (2^24)/7

Now convert we have to convert the numerator to 1 or -1 in relation to denominator.

So convert (2^24)/7 to [(2^3)^21]/7

So that it becomes (8^21)/7

Now when you divide 8 by 7, the remainder is 1.

So we get 1^21 which equals 1.

So the remainder is 1. Ans.

Let us take another example to better understand this Number System - divisibility shortcut trick.

Example 3. What is the remainder when 51^202 is divided by 7.

Here is Easy shortcut for this Number System -  divisibility question.

Divide 51 by 7. The remainder is 2.

So we have (2^202)/7

Which equals [(2^3)^67x (2^1)]/7

Since 2^3 equal 8. Dividing it with 7, we get remainder 1.

So it becomes [(1^67)x2]/7

1^67 equals 1. Multiplying it with 2, we get 2.

So we get 2/7. So the remainder is 2.

Note: When a small numerator is divided by large denominator, the remainder is always that numerator i.e. for 1/7, the remainder is 1.

For 3/7, the remainder is 3. For 5/9, the remainder is 5.