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
2. Divisible by 3
3. Divisible by 4
4. Divisible by 5
5. Divisible by 6
6. Divisible by 7
- 0, or
- divisible by 7
7. Divisible by 8
8. Divisible by 9
9. Divisible by 10
10. Divisible by 11
- 0, or
- divisible by 11
Finding the Unit's Place Digit in a Number in the form of N^{n}
- 0 - Then Unit digit of X^{Y} will be the unit digit of X^{4}
- 1 - Then Unit digit of X^{Y} will be the unit digit of X^{1}
- 2 - Then Unit digit of X^{Y} will be the unit digit of X^{2}
- 3 - Then Unit digit of X^{Y} will be the unit digit of X^{3}
Example:
Find the digit in the unit place of the number 7^{95}
Solution:
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
Question Type 2: Based on Finding the Unit Place of Number
Question Type 3: Based on the Sum of Consecutive Numbers
Question Type 4: Based on Rules of Divisiblity