- Number System
- Algebra
- Area
- Permutation and Combination
- Profit and Loss

Counting numbers are called natural numbers.

Thus 1,2,3,........ all are natural numbers

All Natural numbers and 0 forms the set of whole numbers

Thus 0,1,2,3,........ are the whole numbers.

clearly, every natural number is a whole number. O is whole number which is not a natural number.

All counting numbers, zero and negatives of counting numbers form the set of Integers

Thus ....-3,-2, -1, 0,1,2,3,........ are Integers.

A Natural number divisible by 2 is called an even number.

Thus 0,2,4,6,8,10,12,14,16........ are even numbers.

A counting number not divisible by 2 is called an odd number.

Thus 1,3,5,7,9,11........ are odd numbers.

A number other than 1 is called a prime number if it is divisible only by 1 and itself.

All prime numbers less than 100 are:-

2,3,5,7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,71, 73, 79, 83, 89 and 97.

Total 25 prime numbers within 100.

Check 12

prime numbers less than 12 are 2,3,5,7,11. Clearly none of them divides 137. So that, 137 is a prime number.

Two natural numbers a and b are said to be co-prime if their HCF is 1.

For example:- (2,3), (4,5), (7,9), (8,11) etc. are pairs of co-primes.

A number, other than 1, which is not a prime number is called a composite numbers.

4,6,8,9,12,14..... etc. are composite numbers.

A series of numbers in which each is geater than that which precedes it by 1 is called consecutive numbers.

For Example:- 1,2,3,4,5,6,7,8 or 15,16,17,18 or 101,102,103 etc.

When the numbers are written in fractions, they are known as Rational Numbers. or, The numbers which can be written in the form p/q (where q ≠ 0) are called Rational numbers. They are denoted by Θ

For Example:- 1/2, 3/99, 45/47 etc.

The number which can not be written in the form of p/q are known as irrational number. (Where p and q are integers are q ≠ 0)

For Example:- √3 = 1.7321.... √2=1.4142.... but recurring decimals like 8/3=2.66666..... can be written in the p/q form so they are Rational Numbers.

Real Numbers include both Rational and Irrational numbers

Divisibility by 2

Any number, the last digit of which is either even or 0 is divisible by 2.

For Example:- 2, 4,20,100,112 etc.

Divisibility by 3

If the sum of the digits of a number divisible by 3, the number is also divisible by 3.

For Example:- 123 ≈ 1+2+3=6, and 6 is divisible by 3 hence 123 is also divisible by3.

Divisibility by 4

If the last two digits of a number is divisible by 4, the number is divisible by 4. The number having two or more zeros at the end is also divisible by 4.

For Example:- 526428,5500, 131000 etc.

Divisibility by 25

The same rule as of 4, is applicable to check the divisibility of 25, A number is divisible by 25 if its last two digits are either Zeros or divisible by 25.

Divisibility by 5

If a number ends in 5 or 0, the number is divisible by 5.

For Example:- 1345, 1340, 25, 22220 etc.

Divisibility by 6

If a number is divisible by both 2 and 3, the number is also divisible by 6. so, for a number to be divisible by 6

1. The number shoulb end with an even digit or 0 and

2. The sum of its digits should be divisible by 3.

Any number, the last digit of which is either even or 0 is divisible by 2.

For Example:- 2, 4,20,100,112 etc.

Divisibility by 3

If the sum of the digits of a number divisible by 3, the number is also divisible by 3.

For Example:- 123 ≈ 1+2+3=6, and 6 is divisible by 3 hence 123 is also divisible by3.

Divisibility by 4

If the last two digits of a number is divisible by 4, the number is divisible by 4. The number having two or more zeros at the end is also divisible by 4.

For Example:- 526428,5500, 131000 etc.

Divisibility by 25

The same rule as of 4, is applicable to check the divisibility of 25, A number is divisible by 25 if its last two digits are either Zeros or divisible by 25.

Divisibility by 5

If a number ends in 5 or 0, the number is divisible by 5.

For Example:- 1345, 1340, 25, 22220 etc.

Divisibility by 6

If a number is divisible by both 2 and 3, the number is also divisible by 6. so, for a number to be divisible by 6

1. The number shoulb end with an even digit or 0 and

2. The sum of its digits should be divisible by 3.

Algebra Formulas