**(NUMBER SYSTEM)**

**Face value of digits:** Face value of a digit in a number is the value of the digit whether this digit is at any place.

**Example:** Face value of 9 in 25946 is 9 and face value of 4 is 4.

**Place value:** The place of a digit in any number is the place value of the number.

**Example:** Place value of 9 in 26950 is 900 as there are two digits after 9 so two zeroes will come after 9. The place value of 6 is 6000 as there are three digits after 6 so three zeroes will come after 6.

**Note:** The face value and the place value of 0 are always 0.

**Example:** The face value and the place value of 0 in 50648 are 0.

**Number:** If the digits are placed at unit, tens, hundreds, thousands……digits then we get a term. This term is a number. A number has many digits while a digit is a single digit.

**Natural Numbers:** We start counting from those numbers, are said to be natural numbers. 0 is not included in natural numbers. Natural numbers are infinite.

**Example:** 1, 2, 3, 4, 5, 6, 7………

**Whole Numbers:** If we add 0 in natural numbers then we get whole numbers.

**Example:** 0, 1, 2, 3, 4, 5…….

**Integer Numbers:** If whole numbers are indicated with positive and negative signs then we get integer numbers.

**Example:** …….-4, -3, -2, -1, 0, 1, 2, 3, 4……..

**Composite Numbers:** Those natural numbers which are divided by 1, itself and other number (at least three factors) are said to be composite numbers.

**Example:** 4, 6, 8, 10, 12, 14, 15………

**Prime Numbers:** Those natural numbers which are divided by 1 and itself (only two factors) are said to be prime numbers.

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

**Smallest prime number is 2.**

**2 is a prime number and an even number so 2 is even prime number.**

**1 is neither a composite number nor prime number.**

**Prime numbers from 1 to 25 = 9**

**Prime numbers from 1 to 50 = 15**

**Prime numbers from 1 to 75 = 21**

**Prime numbers from 1 to 100 = 25**

**Prime numbers from 1 to 125 = 30**

**Prime numbers from 1 to 150 = 35**

**Prime numbers from 1 to 175 = 40**

**Prime numbers from 1 to 200 = 46**

**Prime numbers from 1 to 300 = 62**

**Prime numbers from 1 to 400 = 78**

**Prime numbers from 1 to 500 = 95**

**Prime numbers from 1 to 1000 = 168**

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**Co-Prime Numbers:** When HCF of two natural numbers is 1 then both numbers are said to be co-prime numbers.

**Example:** (2, 5) (3, 7) (4, 9)……..etc.

**Even Numbers:** Those numbers are divided by 2 are even numbers.

**Example:** 2, 4, 6, 8, 10, 12……..etc.

**Odd Numbers:** Those numbers are not divided by 2 are odd numbers.

**Example:** 1, 3, 5, 7……..

**Note:** The sum of two even numbers is always even number and the sum of two odd numbers is always even number.

**Rational Numbers:** If an integer is divided by other integer (Except 0) then we get a rational number. So Rational number = p/q where q is not equal to 0.

In decimal fractional numbers, digits could be counted or could be repeated after decimal then these numbers will be rational numbers.

**Example:** 1, 2, 3, 4, ,3, 7, …..etc.

**Irrational Numbers:** Such numbers that can not be written as are said to be irrational numbers.

or those numbers that have not definite.

e and π are irrational numbers.

Such numbers are not perfect square numbers.

**IMPORTANT POINTS: **

**All natural numbers, whole numbers and integers are rational numbers.**

**The product and sum of two rational numbers is always a rational number.**

**The product and sum of a rational number and an irrational number is always an irrational number.**

**Precedent Number:** Just before number from a number.

**Example:** 751 is precedent number of 752.

**Subsequent Number:** Just after number from a number.

**Example:** 16544 is an subsequent number from 16543.

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**(Rule of divisibility)**

**Rule of divisibility by 2:** If a number has 0,2,4,6,8 as its last digit then this number is divided by 2.

**Example:** 4350, 42588, 56744

**Rule of divisibility by 3:** If the sum of the digits of a number is divided by 3 then the number is always divided by 3.

**Example:** 85761, 8 + 5 + 7 + 6 + 1 = 27

Here 27 is divided by 3 so the number will be divided by 3.

**Rule of divisibility by 4:** If the last two digits of a number is divided by 4 then the number will be divided by 4.

**Example:** 15396, Here 96 is divided by 4 then the number will be divided by 4.

**Rule of divisibility by 5:** If the last digit of a number is 0 or 5 then the number will be divided by 5.

**Example:** 85970, 45745

**Rule of divisibility by 6:** If a number is divided by 2 and 3 then the number will be divided by 6.

**Examples:** 5730, 85944

**Rule of divisibility by 7:** The twice of the unit number of the given number is subtracted from the rest number. If the rest number is divided by 7 then the given number will be divided by 7.

**Example:** 16807, twice of 7 (14) is subtracted.

1680 – 7 × 2 = 1666, 166 – 6 × 2 = 154,

15 – 4 × 2 = 7 this is divided by 7.

If the same digit of a number is repeated by 6 times then the formed number is always divided by 7. Example: 444444 is divided by 7.

**Rule of divisibility by 8:** If the last three digits of a number is divided by 8 then the number will be divided by 8.

**Example:** 73584, the last three digits are 584 and 584 is divided by 8 then the given number will be divided by 8.

**Rule of divisibility by 9:** If the sum of the digits of a number is divided by 9 then the number will be divided by 9.

**Example:** 47691, 4 + 7 + 6 + 9 + 1 = 27

27 is divided by 9 then the number will be divided by 9.

**Rule of divisibility by 11:** If the difference of the sum of even placed digits and odd placed digits is divided by 0 or 11 then the number will be divided by 11.

**Example:** 95744,

(9 + 7 + 4) – (5 + 4) = 20 – 9 = 11

Their difference is divided by 11 so the number will be divided by 11.

**Rule of divisibility by 13:** The four times of the unit number of the given number is added in the rest number. If the rest number is divided by 13 then the given number will be divided by 13.

**Example:** 11648, on adding 4 times of the unit number

1164 + 32 = 1196

119 + 24 = 143

14 + 12 = 26

26 is divided by 13 then the number will be divided by 13.

**Rule of divisibility by 17:** The 5 times of the unit number of the given number is subtracted from the rest number. If the rest number is divided by 17 then the given number will be divided by 17.

**Example:** 16779, on subtracting 5 times of the unit number

1677 – 45 = 1632

163 – 10 = 153

153 is divided by 17 then the number will be divided by 17.

**Rule of divisibility by 19:** The twice of the unit number of the given number is added in the rest number. If the rest number is divided by 19 then the given number will be divided by 19.

**Example:** 1862, on adding twice of the unit number

186 + 4 = 190 which is divided by 19.

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**NOTE:** **If a number is formed by writing a digit 6 times then this number is also divided by 3, 7, 11, 13 and 37.**

**Example:** If a number is formed by writing a digit 6 times then from what option the number will be divided?

(1) 7 (2) 13

(3) 37 (4) 19

**Answer – 19**

**Short Trick-1:** **Sum of n natural numbers = {n(n+1)}/2**

**Example:** Sum of natural numbers from 1 to 100.

= {n(n+1)}/2 = {100(101)}/2 = 50(101) = **5050**

**Short Trick-2: ****Sum of consecutive numbers from a to b = {Sum´(Difference + 1)}/2 **

**Example:** Sum of consecutive numbers from 51 to 100.

Here a = 51, b = 100

= {(51 + 100) (100 – 51 + 1)}/2

= (151 *´ 50)/2 = **3775**

**Short Trick – 3:** **Sum of n consecutive even numbers = n(n+1) Where n are even numbers.**

**Example:** Find the sum of even numbers from 1 to 100.

**Answer:** Total even numbers up to 100 = 100/2 = 50 then **Sum = n(n+1) = 50(50 +1) = 50 × 51 = 2550**

**Short Trick – 4: ****Sum of n consecutive odd numbers = n ^{2}**

**Example:** Find the sum of odd numbers from 1 to 100.

**Answer:** Total odd numbers up to 100 = 100/2 = 50

Sum of odd numbers up to 100 **= n ^{2} = 50^{2} = 2500**

**Short Trick – 5 : ****Sum of squares of n natural numbers = n(n+1)(2n+1)/6**

**Example:** Find the sum of the squares of natural numbers from 1 to 10.

**Answer: **1^{2} +2^{2} + 3^{2}+ 4^{2 }+ 5^{2 }+ 6^{2 }+ 7^{2 }+ 8^{2 }+ 9^{2 }+ 10^{2} = ?

Here n =10,

**= {n(n+1)(2n+1)}/6 = {(10 × 11 × 21)}/6 = 385**

**Short Trick – 6 : ****Sum of squares of n odd numbers = {n(4n ^{2} – 1)}/3**

**Example:** 1^{2} + 3^{2} + 5^{2} + ……. + 19^{2} = ?

Where n = 10 are odd numbers.

**= {10(4.10 ^{2} – 1)}/3 = (10 × 399)/3 = 1330**

**OR**

**Short Trick – 7: ****Sum of squares of n even/odd numbers = {n(n+1)(n+2)}/6**

**Example:**

Here n = Total numbers = 19

**= {n(n+1)(n+2)}/6 = (19 × 20 × 21)/6 =1330**

**Short Trick – 8 : ****Sum of cubes of n natural numbers = {n(n+1)/2} ^{2}**

**Example:** Find the sum of cubes of natural numbers from 1 to 7.

**Answer:** 1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} + 6^{3} + 7^{3} = ?

Sum = **{n(n+1)/2} ^{2} = {7 (7+1)/2}^{2} = {(7 × 8)/2}^{2} = 784**

**Short Trick – 9 : ****Sum of n even cube numbers = 2n ^{2 }(n + 1)^{2}**

Here n are even numbers.

**Example:** 2^{3} + 4^{3} + 6^{3} + …….+ 20^{3} = ?

**Answer:** Total numbers n = 20/2 = 10

**= 2n ^{2}(n + 1)^{2 }= 2.10^{2} (10 + 1)^{2} = 2 × 100 × 121 = 24200**

**Short Trick – 10 : ****Sum of n multiples of a number = {xn(n + 1)}/2**

**Example:** Sum of 6 multiples of 2.

**= {xn(n + 1)}/2 = {2.6(6 + 1)}/2 = 42**

**Short Trick – 11 : ****nth term (Last Term) of an A.P (Arithmetic Pregression)**

Tn = a + (n – 1)d

**Example:** 2, 5, 8, 11………12^{th} term = ?

a = 2, n = 12, d = 5 – 2 = 3

**Tn = a + (n – 1)d = 2 + (12 – 1)3 = 2 + 33 = 35**

**Short Trick – 12: ****Sum of n terms of an A.P (Arithmetic Progression) = {n (a + l)} / 2**

Where n = Total terms, a = First term, l = Last term

**Example:** 2 + 5 + 8 + 11………+ 29 = ?

**Answer:** Sum = {n (a + l)} / 2

Tn = l = a + (n – 1) d ( l = 29, a = 2, d = 5 – 2 = 3)

29 = 2 + (n – 1) 3 , n = 10

**Sum = {n (a + l)} / 2 ={10 (2 + 29)} / 2 = 5 × 31 = 155**

**Short Trick – 13 : ****nth term of a G.P (Geometric Progression):**

**Tn = ar ^{n-1}**

**Example:** 3, 6, 12, 24, 48………..10^{th} term = ?

a = 3, r = 6/3 = 2, n = 10

**Tn = ar ^{n-1} = 3(2)^{10-1} = 3 × 2^{9} = 3 × 512 = 1536**

**Important Points about Number System: **

**Total one digit numbers in Number System = 9**

**Total two digits numbers in Number System = 90**

**Total three digits numbers in Number System = 900**

**Total four digits numbers in Number System = 9000**

**Total digits in numbers from 1 to 100 = 192**

**Total zeroes in numbers from 1 to 100 = 11**

**Total 1s(ones) in numbers from 1 to 100 = 21**

**Rest numbers(Except 0 and 1) from 1 to 100 = 20**

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