In competitive examinations like SSC (CGL, CHSL, MTS, CPO), Railway (RRB NTPC, GROUP D, ALP), Bank (RRB PO, CLERK), Lekhpal, UP SI, Delhi Police, etc 1 or 2 questions are asked from the number system (संख्या प्रणाली) so it is important that the students should understand the basic concept of the number system in Maths.

The number system is a very easy topic, in this topic questions are raised on many patterns regarding numbers like the rational number, an irrational number, whole number, even and odd number, a composite number, natural number, real number, unit digit, last unit digit, etc.

In this post, I have told some common tricks to solve the number system questions, which some students may already know. For Practice Download **Number System Questions Pdf** In Hindi and English.** Link is given Below**

#### TYPES OF NUMBERS

**1. REAL NUMBER –** From -Infinite………to………. +infinite

**2. INTEGERS –** -1, -2, -3, -4………+1, +2, +3, +4……

**3. NATURAL NUMBERS –** 1, 2, 3, 4, 5 6……….

**4. WHOLE NUMBERS –** +0, 1, 2, 3, 4, 5, 6, 7……

**5. EVEN NUMBERS –** The Number which is Divided by 2 – 2, 4, 6, 8, 10……

**6. ODD NUMBERS –** The Number which is not complete divided by 2 – 3, 5, 7, 9, 11.…..

**7. PRIME NUMBERS –** The Number which is divided by 1 and itself – 1, 2, 3, 5, 7, 11, 13…..

**8. CO – PRIME NUMBER –** The Numbers whose HCF are not same – 7, 19 – HCF is 1, 23, 27 – HCF is 1

**9. RATIONAL NUMBERS –** The Number which is in Integers and written in P/Q like 5/7, -3/2, 8/11, 17/15, 0/1 is also a rational number but 2.2/3, 5.8/11, 7.2/7 these numbers are not rational numbers.

**10. IRRATIONAL NUMBERS –** The Number which is not a rational number is called an irrational number like 1.333, √3, √5, √8, not repeated

**11. COMPOSITE NUMBERS –** The Number which is divided by 1 or itself and other numbers.

**8 –** 1, 2, 4, 8, **16 –** 1, 2, 4, 8, 16, **27 –** 1, 3, 9, 27

**12. RECURRING DECIMAL (BAR) –** 1.333 = 1.3bar, 2.6555555 = 2.65bar

How to solve recurring decimal = after decimal number – without bar number/ 9 (depend on bars number and 0 without bar numbers)

EX – **1.3bar** = 3 – 0/9 = 3/9 = 1/3 ans

(**ii) 2.536666 = 2.536bar** = 536 – 53/900 = 483/900 = 161/300 = 2+ 161/300 = 761/300 ans

**DIVISIBILITY RULE**

**RULE OF 2, 4, 8, 16, 32**

If the last digit of a number is divided by 2, then the complete number is also divided by 2 = 1578948/2 = 789474

If the last 2 digits of a number is divided by 4, then the complete number is also divided by 4 = 896748/4 = 224187

If the last 3 digits of a number is divided by 8, then the complete number is also divided by 8 = 458656/8 = 57332

If the last 4 digits of a number is divided by 16, then the complete number is also divided by 16

**RULE OF 5 –** Last Digit should be 0 or 5

**RULE OF 6 – **The number which is divided by both 2 and 3

**RULE OF 9 –** Add all the digits of a number and divide by 9 = 89654/9

= 32/9 not divide, 795627/9 = 36/9 = 4 it is divide

**RULE OF 11 –** Sum of the digit at even places – Sum of digits at odd places from right side = 3547124 = 10 – 14 = – 4/11 it is not divided by 11

#### HOW TO FIND UNIT DIGIT

**(I) In Simple Numbers – Take Last Digit and Sign**

Ex – 27981 + 964751 – 14745 + 47458 = 1 + 1 – 5 + 8 = 10 – 5 = 5 unit digit

4756 x 145 x 748 x 1412 = 6 x 5 x 8 x 2 = 30 x 16 = 0 unit digit

5678 x 896 + 4785 – 9874 = 8 x 6 + 5 – 4 = 40 +1 = 41 = unit digit is 1

**(II) Number with Power – Divide Power with 4**

If reminder 1 then the power of a last digit of the number is 1, If the remainder is 2 then the power of a last digit of the number is 2, If reminder 3 then the power of a last digit of the number is 3, and If reminder 0 then the power of a last digit of the number is 4.

Ex – **(314)**^{171}^{}= 171/4 = 4 x 42 =168, reminder is 3 then last digit of

Number is 4^{3 }= 64 = unit digit is 4 answer

**(78)**** ^{1001 }**= 1001/4 = 4x 250, reminder is 1 = (8)

^{1 }= unit digit is 8 answer

##### How to Find Last 2 Digits of a Number

(i) Ending with Odd Numbers – 1, 3, 5, 7, 9

(ii) Ending with Even Numbers – 2, 4, 6, 8, 0

(iii) Ending with 5 – Last 2 Digits of ending with 5 is always be 25.

(iv) Ending with 1 – (261)^{423 }= Take 2^{nd} digit of this number which is 6 and multiply with power’s last digit which is 3 = 6 x 3 = 18, here take last of 18 which is 8 and last digit of question’s number which is 1 so 81 is the last 2 digit of this number.

(v) Ending with 3, 5, 7, 9 – Try to make last digit of all the number is in the form of 1 and divide the power by 4.

Ex – (3)^{243 }= 3^{(4)}^{60} x 3^{3} = (81)^{60} x 27 = 0 x 1 x 27 = Last 2 Digits is 27 ans

Ex – (7)^{325} = 7^{4} x 7 = (2401)^{81} x 7 = 01 x 7 = 07 last 2 digits (325/4 = 4 x 81 + 7^{1})

#### How to find Total, Even and Odd Number of Factors

**TRICK – **Find the Prime Factors, Write down all the powers, Add 1 to each exponent and multiply all Examples –

(1) 40 = 5^{1} x 8^{1} = 5^{1} x 2^{3} = (1 +1) x (3 + 1) = 2 x 4 = Total number of factor is 8

(2) 196 = 2^{2} x 7^{2} = (2 + 1) x (2 + 1) = 3 x 3 = 9 (Here 7 is odd then (2 + 1 = 3) is odd find Even 9 – 3 = 6 numbers are even and 3 are odd)

(3) 120 = 3 x 5 x 8 = 3^{1} x 5^{1} x 2^{3} = (1+1) x (1+1) x (3+1) = 2 x 2 x 4 = 16 Total number of factor 3 and 5 are odds 2 x 2 = 4 odd numbers and 16 – 4 = 12 even numbers

**PRIME FACTORS** – The number which is divided into positive integers is called prime factor. Ex –

(1) 36 – 2 x 2 x 3 x 3 = Total number of Prime factors is 4 and Total numbers of Different are 2 (2, 3).

(2) 305x 277 = 2^{5} x 3^{5} x 5^{5} x 3^{7} x 3^{7} = 15 + 14 = Total 29 and Different 3 (2, 3, 5)

##### Sum of Factors

**(1) **14 – 2^{1} x 7^{1} = (2^{0} + 2^{1}) (7^{0} +7^{1}) = (1+2) (1+7) = 3 x 8 = 24 (Any number with power 0 is always count 1)

(2) 48 – 2x2x2x2x3 = 2^{4} x 3^{1} = (2^{0}+2^{1}+2^{2}+2^{3}+2^{4}) (3^{0}+3^{1}) = 31 x 4 = 124 Sum of 48

##### Product of Factors

(1) 90 = 2^{1} x 3^{2} x 5^{1 }= (1+1) (2+1) (1+1) = 12 Total number divided by 2

12/2 = 6 so, the Product is 90^{6}.

#### Number System Practice Questions

**1.** The sum of the first two of three consecutive odd numbers is 33 more than the third number.

What is the second number?

(1) 35 (2) **37** (3) 39 (4) 33

**2.** Find the number nearest to 77685 which is exactly divisible by 720.

(1) 78680 (2) 77700 (3) **77760** (4) 78960

**3. **What will be the unit’s digit in the value of (3127)173?

(1) 9 (2) 1 (3) 3 (4)** 7**

**4.** Find the number of prime factors in the product of 25^{12 }× 10^{7} × 14^{7}.

(1) 50 (2) **52** (3) 51 (4) 54

**5.** If a number is as much greater than 31 as it is less than 75, then the number is

(1) 106 (2) 44 (3) 74 (4) **53**

**6. **A number when divided by 899 gives a remainder 63. If the same number is divided by 29,

the remainder will be :

(1) 10 (2) **5** (3) 4 (4) 2

**7.** The divisor is 25 times the quotient and 5 times the remainder. If the quotient is 16,

the dividend is :

(1) 6400 (2) **6480 (**3) 400 (4) 480

**8.** When a number is divided by 56, the remainder obtained is 29. What will be the remainder

when the number is divided by 8 ?

(1) 4 (2) **5** (3) 3 (4) 7

**9.** Which of the following number is NOT divisible by 18?

(1) 54036 (2) 50436 (3) 34056 (4) **65043**

**10.** 64329 is divided by a certain number. While dividing, the numbers, 175, 114 and 213 appear

as three successive remainders. The divisor is

(1) 184 (2) 224 (3) **234** (4) 296

**11.** If 17^{200} is divided by 18, the remainder is—

(1) 17 (2) 16 (3) **1** (4) 2

**12. **(4^{61} + 4^{62 }+ 4^{63}) is divisible by

(1) **3** (2) 11 (3) 13 (4) 17

**13. **When an integer K is divided by 3, the remainder is 1, and when K + 1 is divided by 5,

the remainder is 0. Of the following, a possible value of K is

(1) 62 (2) 63 (3) **64** (4) 65

**14**. What least number, of 5 digits is divisible by 41?

(1) 10045 (2) **10004** (3) 10041 (4) 41000

**15.** How many numbers between 1000 and 5000 are exactly divisible by 225 ?

(1)16 (2)**18** (3)19 (4)12

**16.** In a farm there are cows and hens. If heads are counted they are 180,

if legs are counted they are 420. The number of cows in the farm is

(1) 130 (2) 150 (3) 50 (4) **30**

**17.** The least number that should be added to 2055, so that the sum is exactly divisible by 27 is

(1) 28 (2) **24** (3) 27 (4) 31

**18.** The sum of two numbers is 75 and their difference is 25. The product of the two numbers is :

(1) 1350 (2) **1250** (3) 125 (4) 1000

**19**. A tree increases annually by 1/8^{th} of its height. By how much will it increase after 2 years,

if it stands today 64 cm high?

(1) 72 cm (2) 74 cm (3) 75 cm (4) **81 cm**

**20.** The digit in unit’s place of the product (2153)^{167} is :

(1) 1 (2) 3 (3) **7** (4) 9

**21.** The unit digit of the expression 25^{6251} + 36^{528 }+ 73^{54} is

(1) 6 (2) 5 (3) 4 (4) **0**

**22**. The sum of all the 2-digit numbers is :

(1) 4995 (2) 4950 (3) 4945 (4) **4905**

**23.** If the difference of two numbers is 3 and the difference of their squares is 39, then the larger number is

(1) **8** (2) 9 (3) 12 (4) 13

**24**. A man ate 100 grapes in 5 days. Each day, he ate 6 more grapes than those he ate on the earlier day.

How many grapes did he eat on the first day ?

(1) **8** (2) 12 (3) 54 (4) 76

**25.** Mohan gets 3 marks for each correct sum and loses 2 marks for each wrong sum.

He attempts 30 sums and obtains 40 marks. The number of sums solved correctly is :

(1) 15 (2) **20** (3) 25 (4) 10

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