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(1)Natural Numbers:They are counting numbers,the set of natural numbers denoted by N.
N={1,2,3,.....}

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(2)Whole Numbers:When we include "zero" in natural number it is called as whole numbers, the set of whole numbers denoted by W.
W= {0,1,2,3,.....}

(3)Integers:Are whole number (None fractional number),that can be positive,negative or zero and set of integers denoted by Z or I.
Z={....-3,-2,-1,0,1,2,3,....}

(i) Negative integers={....-3,-2-1}
(ii) Positive integers={1,2,3,....}
(iii)Non-negative integers={0,1,2,3....}

Note:Fractional numbers are not integers,
So , 3/5,1/2,4/9,2.21,9.6,3.68 etc.
are not integers.

(4)Prime Numbers:A number other than 1 is known as a prime number if it divisible by 1 and itself only.

Note:(i) Prime number are not divisible by any other numbers, except 1 and itself.
(ii) 1(one) is not a prime number.
(iii) 2 is only even prime number.
(iv) There are 25 prime numbers, between 1 and 100.

(5) Co-Prime: Pair of two natural numbers having no common factor,other than 1,is called a pair of Co-prime.

eg , (6,13), (11,17), (19,21) etc.
Note: H.C.F of co-prime number is 1 (one).

(6) Twin Primes: Prime numbers differing by two are called twin prime numbers.
eg.(11,13), (17,19), (29,31) etc.

(7) Prime Triplet: The set of three consecutive primes is called a prime triplet.
eg. {3,5,7}.

(8)Composite numbers: These numbers have more than two factors(one and itself),are known as composite numbers.

Note: (i) 1(one) is not a composite number.
(ii) Composite number may be even or odd.

(9) Even Numbers:Which are exactly divisible by 2.

(10)Odd Numbers:Which are not exactly divisible by 2.
Note:Zero is neither even nor odd.

(11)Rational Numbers:A number that can be written in the form p/q is called a rational number, where p and q are integers ,q is not equal to zero.
The set of rational numbers denoted by Q.
Note:
(i) Every integer is a rational number.
(ii) A rational number is either an integer or a non-integer.
(iii) Zero is an rational number.
e.g. 0, 1, 2, -3 etc.

(12) Irrational Numbers:A number that can't be written in the form p/q is called an irrational number, where p and q are integers, q is not equal to 0.

Note:(i) Surds are irrational numbers.
(ii) 0(zero)is not an irrational number.
eg.(2)^1/2, 0.30,3000 ....

(13) Real Numbers:The set of all rational and irrational numbers are known as real numbers.
eg. 0, 1/2, (3)^1/2 etc.

Tips on Division

If a number n is divisible by two co-primes numbers a, b then n is divisible by ab.

(a-b) always divides (aⁿ - bⁿ) if n is a natural number.

(a+b) always divides (aⁿ - bⁿ) if n is an even number.

(a+b) always divides (aⁿ + bⁿ) if n is an odd number.

Division Algorithm

When a number is divided by another number then

Dividend = (Divisor x Quotient) + Reminder

Series

Following are formulaes for basic number series:

(1+2+3+...+n) = (1/2)n(n+1) (sum of first n natural numbers)

(1²+2²+3²+...+n²) = (1/6)n(n+1)(2n+1) (sum of square of first n natural numbers)

(1³+2³+3³+...+n³) = (1/4)n²(n+1)²(sum of cube of first n natural numbers)

Basic Important Formulas

These are the basic formulae:

(a + b)² = a² + b² + 2ab

(a - b)² = a² + b² - 2ab

(a + b)² - (a - b)² = 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a² - b²) = (a + b)(a - b)

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

(a³ + b³) = (a + b)(a² - ab + b²)

(a³ - b³) = (a - b)(a² + ab + b²)

(a³ + b³ + c³ - 3abc) = (a + b + c)(a² + b² + c² - ab - bc - ca)

Divisibility by 2:
Any even number or number whose last digit is an even number i.e. 2,4,6,8 including 0 is always completely divisible by 2.

Example: 5206 is an even number and divisible by 2.

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Divisibility by 3:
Divisibility rule for 3 states that a number is completely divisible by 3 if the sum of its digits is divisible by 3 i.e., it is a multiple of 3

Consider a number 432, in this number there are three digits and sum of three digits 4+3+2 is 9 that is divisible by 3, so number 432 will also be divisible by 3.

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Divisibility by 4:
If the last two digits of a number are divisible by 4, then that number is a multiple of 4 and is divisible by 4 completely.

Example: Take a number 4328. Consider the last two digits i.e. 28. As 28 is divisible by 4, the original number 4328 is also divisible by 4.

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Divisibility by 5:
Numbers with last digit 0 or 5 are always divisible by 5.
Example: 10, 10000, 60000005, 12595, 3963454850 etc.

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Divisibility by 6:
Numbers which are divisible by both 2 and 3 are divisible by 6. T hat is, if last digit of the given number is even and the sum of its digits is a multiple of 3, then the given number is also a multiple of 6.

Example: 714, the number is divisible by 2 as the last digit is 4.
The sum of digits is 7+1+4 = 12, which is also divisible by 3.
so 714 is divisible by 6.

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Divisibility by 7:
The rule for divisibility by 7 is given below:

Example: Is 1073 divisible by 7?

• From the rule stated remove 3 from the number and double it, which becomes 6.
• Remaining number becomes 107, so 107-6 = 101.
• Repeating the process one more times, we have \(1 \times 2 = 2\)
• Remaining number 10 – 2 = 8.
• As 8 is not divisible by 7, hence the number 1073 is not divisible by 7.

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Divisibility by 8:
If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.

Example: Take a number 54784. Consider the last three digits i.e. 784. As 784 is divisible by 8, the original number 54784 is also divisible by 8.

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Divisibility by 9:
Rule for divisibility by 9 is similar to divisibility rule for 3. That is, if the sum of digits of the number is divisible by 9, then the number itself is divisible by 9. Example: Consider 68652, as the sum of its digits (6+8+6+5+2) is 27, which is divisible by 9, hence 68652 is divisible by 9

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Divisibility by 10:
Divisibility rule for 10 states that any number whose last digit is 0, is divisible by 10.

Example: 10, 20,30,1500,7000,645000 etc.

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Divisibility by 11:
If the difference of the sum of alternative digits of a number is divisible by 11 then that number is divisible by 11 completely.

In order to check whether a number like 924165 is divisible by 11, because the difference of 9+4+6=19 and 2+1+5=8 is 11 that is why the number 924165 is divisible by 11.

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Divisibility by 12:

Rule : Any number which is divisible by both 4 and 3, is also divisible by 12.

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Divisibility by 13:

Rule : Oscuator for 13 is 4 (see note). But time, our osculator  is not negative (as in case of 7). It is ‘one-more’ osculator. So, the working principle will be different now. This can be seen in the following examples.

Ex 1: Is 143 divisible by 13?

Solution : 14 3 : 14 + 3 × 4 = 26

Divisibility by 14:

Any numberwhich is divisible by both 2 and 7, in also divisible by 14. That is, the number’s last digit should be even and at the same time the number should be divisible by 7.

Divisibility by 15:

Any number which is divisible by both 3 and 5 is also divisible by 15.

Divisibility by 16:

Any number whose last 4 digit number is divisible by 16 is also divisible by 16.

Divisibility by 17:

Negative osculator for 17 is 5 (see note). The working for this is the same as in the case of 7.

Ex1: cheek the divisible of 1904 by 17.

Solution :

190 4 : 190 – 5 × 4 = 170

Since 170 is divisible by 17, the given number is also divisible by 17.

Note : students are suggested not to go upto the last calculation. Whenever you find the number divisible by the given number on right side of your calculation stop further calculation and conclude the result.

Divisibility by 18:

Rule : Any number which is divisible by 9 and has its last digit even or zero, is divisible by 18.

Ex1: 926568 : digit -sum  is a multiple of nine (i.e. divisible by 9) and unit digit(8) is even, hence the number is divisible by 18.

Divisibility by 19:

If you recall, the ‘one- more’ osculator for 19 is 2. The method is similar to that of 13, which is well known to you. Let us take an example,

Ex1 : 149264

Solution : 1 4 9 2 6 4

19 / 9 / 12 / 11 / 14

Thus, our number is divisible by 19.

Note: you must have understood the working principle (see the case of 13).

We hope that the post would have cleared all your doubts related to the topic.