Number System

Important Terms

Natural Number N = { 1, 2, 3, 4, 5, ......  }
Whole Number W = { 0, 1, 2, 3, 4, 5,........  }
Integer I = { -∞ ...., -3, -2, -1, 0, 1, 2, 3, 4......  }
Positive Integers I+ = { 1, 2, 3, 4, 5, ..........  }
Negative Integers I- = { -1, -2, -3, -4 ....... -∞ }
Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, p and q, with the denominator q not equal to zero.

eg. 3/2, -5 as it can be written as -5/1 , 1.5 as it can be written as 3/2

Note:

All Natural Numbers, All Whole Numbers & All Integers are rational Numbers.

Irrational Numbers

Those numbers which cannot be expressed in the form of p/q, where q≠0.

eg. √ 2, √ 3, √ 5 , π etc are irrational numbers

Real Numbers  
Even Numbers

Numbers which are divisible by 2

eg. 2, 4, 6, 8...... are even numbers

Odd Numbers

Numbers which are not exactly divisible by 2 

eg. 1, 3, 5, 7.... are odd numbers

Prime Numbers

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

eg.  2, 3, 5, 7 etc

Note:

 2 is the only Even number which is Prime. All other Prime Number are Odd.

Composite Numbers

A natural number greater than 1 that is not a prime number is called a composite number. Means a composite number has other other factors including itself and unity.

Note:

1 is neither prime nor Odd.

Composite Number can be even or odd.

Perfect Numbers

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum)

eg. 6 is perfect number because 1, 2, and 3 are its proper positive divisors, and sum of its proper positive divisors = 1 + 2 + 3 = 6.

 28 is perfect number because 1, 2, 4, 7, 14 are its proper positive divisors and sum of its proper positive divisors = 1 + 2 + 4 + 7 + 14 =28

Note:

The Sum of Reciprocal of the divisors of a perfect number including that of its own is always equal to 2. 

eg. 1, 2, 3 are the proper divisors of 6. Now 1/1 +1/2 + 1/3 + 1/6 = 2 

Decimal Numbers A Decimal Number is a number that contains a Decimal Point. eg. 0.1, 1.2, 127.5

 

Test of Divisibility

1. Divisible by 2

A number is divisible by 2  if the last digit is 0, 2, 4, 6 or 8.
Example:
68 is divisible by 2 since the last digit is 8.
 

2. Divisible by 3

A number is divisible by 3  if the sum of the digits is divisible by 3.
Example:
168 is divisible by 3 since the sum of the digits is 15 (1+6+8=15), and 15 is divisible by 3.
 

3. Divisible by 4

A number is divisible by 4  if the number formed by the last two digits is divisible by 4.
Example:
516 is divisible by 4 since 16 is divisible by 4.
 

4. Divisible by 5

A number is divisible by 5  if the last digit is either 0 or 5.
Example:
535 is divisible by 5 since the last digit is 5.

 

5. Divisible by 6

A number is divisible by 6  if it is divisible by 2 AND it is divisible by 3.
Example:
36 is divisible by 6 since it is divisible by 2 and it is divisible by 3 also.
 

6. Divisible by 7

If you double the last digit and subtract it from the rest of the number and the answer is:
  • 0, or
  • divisible by 7
(Note: you can apply this rule to that answer again if you want)
 
Example:
672 (Double 2 is 4, 67-4=63, and 63÷7=9) Yes
905 (Double 5 is 10, 90-10=80, and 80÷7=11 3/7) No

 

7. Divisible by 8

A number is divisible by 8  if the number formed by the last three digits is divisible by 8.
Example:
7120 is divisible by 8 since 120 is divisible by 8.
 

8. Divisible by 9

A number is divisible by 9  if the sum of the digits is divisible by 9.
Example:
549 is divisible by 9 since the sum of the digits is 18 (5+4+9=18), and 18 is divisible by 9.
 

9. Divisible by 10

A number is divisible by 10  if the last digit is 0.
Example:
390 is divisible by 10 since the last digit is 0.

 

10. Divisible by 11

If you sum every second digit and then subtract all other digits and the answer is:
  • 0, or
  • divisible by 11
 
Example:
1364 ((3+4) - (1+6) = 0) Yes
3729 ((7+9) - (3+2) = 11) Yes
25176 ((5+7) - (2+1+6) = 3) No

 

Finding the Unit's Place Digit in a Number in the form of Nn

Let XY is the number
 
Steps to Find the Unit Digit:
STEP 1: Divide the Y by 4 and find the Remainder.
STEP 2: If the Remainder is:
  • 0 - Then Unit digit of XY will be the unit digit of X4
  • 1 - Then Unit digit of XY will be the unit digit of  X1
  • 2 - Then Unit digit of XY will be the unit digit of  X2
  • 3 - Then Unit digit of XY will be the unit digit of X3

Example:

Find the digit in the unit place of the number 795

Solution:

Divide 95 by 4: the remainder is 3.
Thus, the last digit of 795 is equals to the last digit of 73=343 i.e. 3.

 

Sum of First n Natural Numbers

Sum of First n even natural Numbers

Sum of First even natural numbers upto n

Sum of First n odd natural numbers 

Sum of Squares of First n natural numbers

Sum of cubes of first n natural numbers

Questions Asked from this Chapter:

 

Question Type 1: Based on Division, Multiplication, Addition and Substraction

 

The sum of three consecutive odd natural numbers each divisible by 3 is 72. What is the largest among them?
a) 21 
b) 24
c) 27
d) 36
 
The product of two positive numbers is 11520 and their quotient is 9/5. Find the difference of two numbers:
a) 60
b) 64
c) 74
d) 70
 
A number being divided by 52 gives remainder 45. If the number is being divided by 13, the remainder will be:
a) 5 
b) 6
c) 12
d) 7

Question Type 2: Based on Finding the Unit Place of Number

 

What will be the unit digit in the product of 7105?
a) 5 
b) 7
c) 9
d) 1
 
One's digit of the number (22)23 is:
a) 4 
b) 6
c) 8
d) 2

 

Question Type 3: Based on the Sum of Consecutive Numbers

 

The sum of all natural numbers from 75 to 97 is:
a) 1598 
b) 1798
c) 1958
d) 1978
 
The sum of all the 2-digit numbers is:
a) 4995
b) 4950
c) 4945
d) 4905
 

Question Type 4: Based on Rules of Divisiblity