# Quantitative Aptitude Number Systems PDF Notes

**Number Systems for CAT | MAT | PSC’s**

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**Natural numbers:
**Counting numbers 1, 2, 3, 4, 5 ……….…… are known as natural numbers

**Whole numbers:**

If we include zero among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5 ……………….. are called whole numbers

**Rational Numbers:
**The numbers of the form x⁄y where X & Y are integers and Y ≠ 0 are known as rational numbers

Example: 2/3, 5/7, -4/9 , etc.

**Irrational/l numbers:
**Those numbers which when expressed in decimal form are neither terminating nor repeating decimals are known as irrational numbers.

Examples: √2, √3, √5, π etc

**Composite numbers:
**Natural numbers greater 1 which are not prime numbers are composite numbers

Example: 4, 6, 9, 15 etc.

**Co – prime numbers:
**Two numbers that have only 1 as the common factors are called co-prime numbers or relatively prime to each other.

Example: (3, 7), (8, 9), (36, 25) etc.

**Learn the Tables from 11 to 20**

11 × 2 = 22 12 × 2 = 24 13 × 2 = 26

11 × 3 = 33 12 × 3 = 36 13 × 3 = 39

11 × 4 = 44 12 × 4 = 48 13 × 4 = 52

11 × 5 = 55 12 × 5 = 60 13 × 5 = 65

11 × 6 = 66 12 × 6 = 72 13 × 6 = 78

11 × 7 = 77 12 × 7 = 84 13 × 7 = 91

11 × 8 = 88 12 × 8 = 96 13 × 8 = 104

11 × 9 = 99 12 × 9 = 108 13 × 9 = 117

11 × 10 = 110 12 × 10 = 120 13 × 10 = 130

11 × 11 = 121 12 × 11 = 132 13 × 11 = 143

11 × 12 = 132 12 × 12 = 144 13 × 12 = 156

11 × 13 = 143 12 × 13 = 156 13 × 13 = 169

11 × 14 = 154 12 × 14 = 168 13 × 14 = 182

11 × 15 = 165 12 × 15 = 180 13 × 15 = 195

11 × 16 = 176 12 × 16 = 192 13 × 16 = 208

11 × 17 = 187 12 × 17 = 204 13 × 17 = 221

11 × 18 = 198 12 × 18 = 216 13 × 18 = 234

11 × 19 = 209 12 × 19 = 228 13 × 19 = 247

11 × 20 = 220 12 × 20 = 240 13 × 20 = 260

14 × 1 = 14 15 × 1 = 15 16 × 1 = 16

14 × 2 = 28 15 × 2 = 30 16 × 2 = 32

14 × 3 = 42 15 × 3 = 45 16 × 3 = 48

14 × 4 = 56 15 × 4 = 60 16 × 4 = 64

14 × 5 = 70 15 × 5 = 75 16 × 5 = 80

14 × 6 = 84 15 × 6 = 90 16 × 6 = 96

14 × 7 = 98 15 × 7 = 105 16 × 7 = 112

14 × 8 = 112 15 × 8 = 120 16 × 8 = 128

14 × 9 = 126 15 × 9 = 135 16 × 9 = 144

14 × 10 = 140 15 × 10 = 150 16 × 10 = 160

14 × 11 = 154 15 × 11 = 165 16 × 11 = 176

14 × 12 = 168 15 × 12 = 180 16 × 12 = 192

14 × 13 = 182 15 × 13 = 195 16 × 13 = 208

14 × 14 = 196 15 × 14 = 210 16 × 14 = 224

14 × 15 = 210 15 × 15 = 225 16 × 15 = 240

14 × 16 = 224 15 × 16 = 240 16 × 16 = 256

14 × 17 = 238 15 × 17 = 255 16 × 17 = 272

14 × 18 = 252 15 × 18 = 270 16 × 18 = 288

14 × 19 = 266 15 × 19 = 285 16 × 19 = 304

14 × 20 = 280 15 × 20 = 300 16 × 20 = 320

17 × 1 = 17 18 × 1 = 18 19 × 1 = 19

17 × 2 = 34 18 × 2 = 36 19 × 2 = 38

17 × 3 = 51 18 × 3 = 54 19 × 3 = 57

17 × 4 = 68 18 × 4 = 72 19 × 4 = 76

17 × 5 = 85 18 × 5 = 90 19 × 5 = 95

17 × 6 = 102 18 × 6 = 108 19 × 6 = 114

17 × 7 = 119 18 × 7 = 126 19 × 7 = 133

17 × 8 = 136 18 × 8 = 144 19 × 8 = 152

17 × 9 = 153 18 × 9 = 162 19 × 9 = 171

17 × 10 = 170 18 × 10 = 180 19 × 10 = 190

17 × 11 = 187 18 × 11 = 198 19 × 11 = 209

17 × 12 = 204 18 × 12 = 216 19 × 12 = 228

17 × 13 = 221 18 × 13 = 234 19 × 13 = 247

17 × 14 = 238 18 × 14 = 252 19 × 14 = 266

17 × 15 = 255 18 × 15 = 270 19 × 15 = 285

17 × 16 = 272 18 × 16 = 288 19 × 16 = 304

17 × 17 = 289 18 × 17 = 306 19 × 17 = 323

17 × 18 = 306 18 × 18 = 324 19 × 18 = 342

17 × 19 = 323 18 × 19 = 342 19 × 19 = 361

17 × 20 = 340 18 × 20 = 360 19 × 20 = 380

**Learn the Square of a numbers from 1 to 99:**

01^{2} = 01 51^{2} = 2601, 11^{2} = 121 61^{2} = 3721

02^{2} = 04 52^{2} = 2704, 12^{2} = 144 62^{2} = 3844

03^{2} = 09 53^{2} = 2809, 13^{2} = 169 63^{2} = 3969

04^{2} = 16 54^{2} = 2916, 14^{2} = 196 64^{2} = 4096

05^{2} = 25 55^{2} = 3025, 15^{2} = 225 65^{2} = 4225

06^{2} = 36 56^{2} = 3136, 16^{2} = 256 66^{2} = 4356

07^{2} = 49 57^{2} = 3249, 17^{2} = 289 67^{2} = 4489

08^{2} = 64 58^{2} = 3364, 18^{2} = 324 68^{2} = 4624

09^{2} = 81 59^{2} = 3481, 19^{2} = 361 69^{2} = 4761

21^{2} = 441 71^{2} = 5041, 31^{2} = 961 81^{2} = 6561

22^{2} = 484 72^{2} = 5184, 32^{2} = 1024 82^{2} = 6724

23^{2} = 529 73^{2} = 5329, 33^{2} = 1089 83^{2} = 6889

24^{2} = 576 74^{2} = 5476, 34^{2} = 1156 84^{2} = 7056

25^{2} = 625 75^{2} = 5625, 35^{2} = 1225 85^{2} = 7225

26^{2} = 676 76^{2} = 5776, 36^{2} = 1296 86^{2} = 7396

27^{2} = 729 77^{2} = 5929, 37^{2} = 1369 87^{2} = 7569

28^{2} = 784 78^{2} = 6084, 38^{2} = 1444 88^{2} = 7744

29^{2} = 841 79^{2} = 6241, 39^{2} = 1521 89^{2} = 7921

41^{2} = 1681 91^{2} = 8281 46^{2} = 2116 96^{2} = 9216

42^{2} = 1764 92^{2} = 8464 47^{2} = 2209 97^{2} = 9409

43^{2} = 1849 93^{2} = 8649 48^{2} = 2304 98^{2} = 9604

44^{2} = 1936 94^{2} = 8836 49^{2} = 2401 99^{2} = 9801

45^{2} = 2025 95^{2} = 9025

**Learn the Cube of a numbers from 1 to 30**

1^{3} = 1 11^{3} = 1331 21^{3} = 9261

2^{3} = 8 12^{3} = 1728 22^{3} = 10648

3^{3} = 27 13^{3} = 2197 23^{3} = 12167

4^{3} = 64 14^{3} = 2744 24^{3} = 13824

5^{3} = 125 15^{3} = 3375 25^{3} = 15625

6^{3} = 216 16^{3} = 4096 26^{3} = 17576

7^{3} = 343 17^{3} = 4913 27^{3} = 19683

8^{3} = 512 18^{3} = 5832 28^{3} = 21952

9^{3} = 729 19^{3} = 6859 29^{3} = 24389

10^{3} = 1000 20^{3} = 8000 30^{3} = 27000

**Find out square when unit digit is 5 ?**

** Q. ****25 ^{2} = ?**

25^{2} = 625

= (2×3) (5^{2})

= 6 25

= 625 (answer)

** Q. 35 ^{2} = ?**

35^{2} = 1225

= (3×4) (5^{2})

= 12 25

= 1225 (answer)

**Q. 45 ^{2} = ?**

45^{2} = 2025

= (4×5) (5^{2})

= 20 25

= 2025 (answer)

** Q. 85 ^{2} = ?**

85^{2} = 7225

= (8×9) (5^{2})

= 72 25

= 7225 (answer)

** Q. 95 ^{2} = ?**

95^{2} = 9025

= (9×10) (5^{2})

= 90 25

= 9025 (answer)

**Find out a square root of a number ?**

**Q. Square root of 8464= ?**

**Sol**. Square root of 8464= 92

8464 ? 8100

> 90^{2}

= 92^{2}

Therefore, Square root of 8464 = 92

**Q. Square root of 9409= ?**

**Sol.** Square root of 9409= 97

9409 ? 9025 (we know that, 95^{2} = 9025)

> 95^{2}

= 97^{
}Therefore, Square root of 9409 = 97 (answer)

**Find out a square of a number ?**

**Q. (106) ^{2} =?**

**Sol.** (106)^{2} = 11236

Step 1: (06)^{2} = 36

Step 2: 1× 2× 06 = 12

Step 3: 1^{2} = 1

Final answer = 1 12 36

(Answers from step 3,step 2,step 1)

= 11236

Therefore, (106)^{2} = 11236 (answer)

** Q. (113) ^{2} = ?**

**Sol.** (113)^{2} = 12769

Step 1: (13)^{2} = 69 (here, 1 is parity or excess)

Step 2: 1× 2× 13 = 26

= 26 + 1 (parity is added here)

= 27

Step 3: 1^{2} = 1

Final answer = 1 27 69

(Answers from step 3,step 2,step 1)

= 12769

Therefore, (113)^{2 } = 12769 (answer)

** Q. (209) ^{2} = ?**

**Sol.** (209)^{2} = 43681

Step 1: (09)^{2} = 81

Step 2: 2 × 2 × 09 = 36

Step 3: 2^{2} = 4

Final answer = 4 36 81

(Answers from step 3,step 2,step 1)

= 43681

Therefore, (209)^{2 } = 43681 (answer)

** Q. (216) ^{2} = ?**

**Sol.** (216)^{2} = 46656

Step 1: (16)^{2} = 56 (here 2 is parity or excess)

Step 2: 2 × 2 × 16 = 64

= 64 + 2 (parity 2 is added here)

= 66

Step 3: ( 2^{2} ) = 4

Final answer = 4 66 56

(Answers from step 3,step 2,step 1)

= 46656

Therefore, (216)^{2} = 46656 (answer)

**To find out a cube root of a number **

**Q. Cubeth root of 1728 ?**

**Sol.** Step 1: in 1728, 8 replaces by 2 (8^{3} = 512)

Therefore, unit place digit is = 2

Step 2: in 1728, ignore 728 (last 3 – digits)

We now have 1

Here, 1? 1^{3} (1 ? 2^{3} )

Therefore, tenth place digit is = 1

Therefore,

Cubeth root of 1728 = 12 (answer)

**Q. Cubeth root of 19683 = ?**

**Sol. **Step 1: in 19683, 3 replaces by 7 (3^{3} = 27)

Therefore, unit place digit is = 7

Step 2: in 19683, ignore 683 (last 3 – digits)

We now have 19

Here, 19 ? 2^{3} (19 ? 3^{3} )

Therefore, tenth place digit is = 2

Therefore,

Cubeth root of 19683 = 27 (answer)

**TO FIND OUT A CUBE OF A NUMBER? **

**TYPE 1: Number starts with ‘1’ (from left)**

Example: 12, 13, 14, 15, ………………….

**Q. (12) ^{3} = ?**

**Sol.** Step 1: write given number as it as with some space

1 2

Step 2: Square and Cube the unit digit (2) of a given

number (12) and write right side to 1 2

1 2 2^{2 }2^{3}

= 1 2 4 8

Step 3: Double the middle numbers (2 & 4 only) and

add to them in the same position

1 2×2 4×2 8

= 1 4 8 8

1 2 4 8

add here 4 8

——————————————-

1 7 (1)2 8 [here, parity or excess ‘1’ added to next left column]

1 7 2 8

So, (12)^{3} = 1728 answer)

**Q.** (13)^{3} = ?

**Sol. **Step 1: write a given number as it as with some space

1 3

Step 2: Square and Cube the unit digit (3) of a given

number (13) and write right side to 1 3

1 3 3^{2 }3^{3}

= 1 3 9 27

Step 3: Double the middle numbers (3 & 9 only) and

add to them in the same position

1 3×2 9×2 27

= 1 6 18 27

1 3 9 27

add here 6 18

—————————————————-

2 (1)1 (2)9 (2)7 [here, parity or excess 2, 2 & 1 shown in the brackets added to next left columns respectively]

2 1 9 7

So, (13)^{3} = 2197 answer)

**Q. (15) ^{3} = ?**

**Sol. **Step 1: write given number as it as with some space

1 5

Step 2: Square and Cube the unit digit (5) of a given

number (15) and write right side to 1 5

1 5 5^{2} 5^{3}

= 1 5 25 125

Step 3: Double the middle numbers (5 & 25 only) and

add them in the same position

1 5×2 25×2 125

= 1 10 50 125

1 5 25 125

add here 10 50

———————————————————

3 (2)3 (8)7 (12)5 [here, parity or excess 12, 8 & 2 shown in the bracket added to next left columns respectively]

3 3 7 5

So, (15)^{3} = 3375 answer)

**TYPE 2: Number ends with ‘1’ (from right)**

Example: 21, 31, 41, 51 …………………….

**Q. (21) ^{3} = ?**

**Sol.** Step 1: write given number as it as with some space

2 1

Step 2: Square and Cube the 10^{th} place digit (2) of a given number (21) and write left side to 2 1

2^{3} 2^{2} 2 1

= 8 4 2 1

Step 3: Double the middle numbers (4 & 2 only) and

add to them in the same position

8 4×2 2×2 1

= 8 8 4 1

8 4 2 1

add here 8 4

————————————————–

9 (1)2 6 1 [here, parity or excess ‘1’ shown in the bracket added to next left columns respectively]

9 2 6 1

So, (21)^{3} = 9261 answer)

**Q. (41) ^{3} = ?**

**Sol.**Step 1: write given number as it as with some space

4 1

Step 2: Square and Cube the 10^{th} place digit (4) of a given number (41) and write left side to 4 1

4^{3} 4^{2} 4 1

= 64 16 4 1

Step 3: Double the middle numbers (16 & 4 only) and

add them in the same position

64 16×2 4×2 1

= 64 32 8 1

64 16 4 1

add here 32 8

—————————————————-

68 (4)9 (1)2 1 [here, parity or excess 1 & 4 shown in the brackets added to next left columns respectively]

68 9 2 1

So, (41)^{3} = 68921 answer)

**TYPE 3: If both the numbers same**

Example: 22, 33, 44, 55 …………………..

**Q. (22) ^{3} = ?**

**Sol.**Step 1: Here, common number is 2

So, write cube of 2 (i. e 2^{3} = 8) in four places with spaces

8 8 8 8

Step 2: Double the middle numbers (8 & 8) add them to the same position

8 8×2 8×2 8

= 16 16

8 8 8 8

add here 16 16

—————————————————–

10 (2)6 (2)4 8 [here, parity or excess 2 & 2 shown in the brackets added to next left columns respectively]

10 6 4 8

So, (22)^{3} = 10648

**Q. (33) ^{3} = ?**

**Sol.**Step 1: Here, common number is 3

So, write cube of 3 (i. e 3^{3} = 27) in four places with some spaces

27 27 27 27

Step 2: Double the middle numbers (27 & 27) add them to the same position

27 27×2 27×2 27

= 54 54

27 27 27 27

add here 54 54

————————————————————

35 (8)9 (8)3 (2)7 [here, parity or excess 2, 8 & 8 shown in the brackets added to next left columns respectively

35 9 3 7

So, (33)^{3} = 35937

**TYPE 4: If both the numbers are different**

Example: 23, 42, 52, 47, 89, ………………………

**(****Q. ****23) ^{3} = ?**

**Sol.**Step 1:write the cube of both the numbers 2 & 3 (i. e 2^{3} = 8 & 3^{3} = 27) with the some space

8 27

Step 2: in the middle 8 & 27, write like 2^{2} × 3 & 3^{2} × 2

8 2^{2} × 3 3^{2} × 2 27

= 8 12 18 27

Step 3: Double the middle numbers (12 & 18) add them to the same position

8 12×2 18×2 27

= 24 36

8 12 18 27

add here 24 36

———————————————————-

12 (4)1 (5)6 (2)7 [Here, parity or excess 2, 5 & 4 shown in the brackets added to next left columns respectively]

12 1 6 7

So, (23)^{3} = 12167

**(35) ^{3} = ?**

**Sol.**Step 1: write the cube of both the numbers 3 & 5 (i. e 3^{3} = 27 & 5^{3} = 125) with the some space

27 125

Step 2: in the middle 27 & 125, write like 3^{2} × 5 & 5^{2} × 3

27 3^{2} × 5 5^{2} × 3 125

= 27 45 75 125

Step 3: Double the middle numbers (45 & 75) add them to the same position

27 45×2 75×2 125

= 90 150

27 45 75 125

add here 90 150

——————————————————–

42 (15)8 (23)7 (12)5 [Here, parity or excess 12, 23 & 15 shown in the brackets added to next left columns respectively]

42 8 7 5

So, (35)^{3} = 42875

**WHEN SUM OF THE UNIT DIGIT IS ‘10’**

**1. 56 × 54 = ?**

Step 1: Multiplication of unit digits i.e 6 × 4

Step 2: Multiplication of 10’s digit, here 5 and its next number i.e 5 × 6

(5 × 6) (6 × 4)

30 24

= 3024 (answer)

**2. 72 × 78 = ?**

Step 1: Multiplication of unit digits i. e 2 × 8

Step 2: Multiplication of 10’s digit, here 7 and its next number i. e 7 × 8

(7 × 8) (2 × 8)

56 16

= 5616 (answer)

**3. 113 × 117 = ?**

Step 1: Multiplication of unit digits i. e 3 × 7

Step 2: Multiplication of 10’s digits, here 11 and its next number i. e 11 × 12

(11 × 12) (3 × 7)

132 21

= 13221 (answer)

**ANY TWO DIGIT MULTIPLICATION**

**1. 42 × 36 = ?**

42 × 36 = ____ ____ ____

Step 1: Multiplication of unit digits ____ ____ 2 × 6

____ ____ 12

Step 2: Sum of multiplication of Extreme numbers and middle numbers i. e (4 × 6) + (2 × 3) = 30

____ 30 12

Step 3: Multiplication of 10’s digits, i. e (4 × 3) = 12

12 30 12

In the above, from right, 1 is treated as excess or parity and it has to added to next number 30

Now, 12 31 2

From the above, 3 is treated as excess or parity and it has to added to next number 12

Now, 15 3 2

= 1532 (answer)

**2. 96 × 73 = ?**

96 × 73 = ____ ____ ____

Step 1: Multiplication of unit digits ____ ____ 6 × 3

____ ____ 18

Step 2: Sum of multiplication of Extreme numbers and middle numbers i. e (9 × 3) + (6 × 7) = 69

____ 69 18

Step 3: Multiplication of 10’s digits, i. e (9 × 7) = 63

63 69 18

In the above, from right, 1 is treated as excess or parity and it has to added to next number 69

Now, 63 70 2

From the above, 7 is treated as excess or parity and it has to added to next number 63

Now, 70 0 8

= 7008 (answer)

**HOW TO LEARN & WRITE TABLES FROM 11 TO 99:**

**Table of 12**

12

To write a table of 12, just write the tables of 1 & 2 in two separate columns

1 2 = 12

2 4 = 24

3 6 = 36

4 8 = 48

5 10 (here, 10’s place digit is added to 5)

(5 + 1) 0 = 60

6 12 = 72 [6 + 1 = 7]

7 14 = 84 [7 + 1 = 8]

8 16 = 96 [8 + 1 = 9]

9 18 = 108 [9 + 1 = 10]

10 20 = 120 [10 + 2 = 12]

**Table of 26
**26

To write a table of 26, just write the tables of 2 & 6 in two separate columns

2 6 = 26

4 12 = 52

6 18 = 78

8 24 (here, 10’s place digit is added to 8

(8 + 2) 4 = 104

10 30 (here, 10’s place digit is added to 10)

(10 + 3) 0 = 130

12 36 = 156 [12 + 3 = 15]

14 42 = 182 [14 + 4 = 18]

16 48 = 208 [16 + 4 = 20]

18 54 = 234 [18 + 5 = 23]

20 60 = 260 [20 + 6 = 26]

**Table of 94
**94

To write a table of 12, just write the tables of 1 & 2 in two separate columns

9 4 = 94

18 8 = 188

27 12 (here, 10’s place digit is added to 27)

(27 + 1) 2 = 282

36 16 = 376 [36 + 1 = 37]

45 20 = 470 [45 + 2 = 47]

54 24 = 564 [54 + 2 = 56]

63 28 = 658 [63 + 2 = 65]

72 32 = 752 [72 + 3 = 75]

81 36 = 846 [81 + 3 = 84]

90 40 = 940 [90 + 4 = 94]

**DIFFERENCE IS ‘10’ AND UNIT DIGIT OR ENDS WITH 5:**

When the difference of given numbers is 10 and unit digit is 5, the number 75 is come right side as common

**1. 35 × 45 = ?**

Step 1: The number 75 is come right side ______ 75

Step 2: The left side number is, square the larger number and substract the ‘1’, i. e (4^{2} – 1) = 15

= 15 75

= 1575 (answer)

**2. 75 × 85 = ?**

Step 1: The number 75 is comes right side _____ 75

Step 2: The left side number is, square the larger number and substract the ‘1’, i. e (8^{2} – 1) = 63

= 63 75

= 6375 (answer)

**3. 135 × 145 = ?**

Step 1: The number 75 is comes right side _____ 75

Step 2: The left side number is, square the larger number and substract the ‘1’, i. e (14^{2} – 1) = 195

= 195 75

= 19575 (answer)

**SAME NUMBERS AND ENDS WITH ‘5’:**

In this case, the number 25 is comes right side as common

**1. 65 × 65 = ?**

Step 1: The number 25 comes right side _____ 25

Step 2: Multiplication of 10’s place digit and its next number i. e (6 × 7) = 42

= 42 25

= 4225 (answer)

**2. 125 × 125 = ?**

Step 1: The number 25 comes right side _____ 25

Step 2: Multiplication of 10’s place digit and its next number i. e (12 × 13) = 156

= 156 25

= 15625 (answer)

**ANY NUMBER AND ENDS WITH ‘5’:**

In this case, when sum of 10’s digits is even number ‘25’ is taken as unit digit (write right side) in the final answer

When sum of 10’s digits is odd number, ‘75’ taken as unit digit (write right side) in the final answer

**1. 45 × 65 = ?**

Step 1: Here, sum of 10’s digit is even (6 + 4 = 10), So, 25 write the right side

_____ 25

Step 2: Multiplication of 10’s digit i. e 4 × 6 = 24

And half the sum of the 10’s digit i. e = 5

Step 3: add the 24 and 5 => 24 + 5 = 29

= 29 25

= 2925 (answer)

**2. 95 × 75 = ?**

Step 1: Here, sum of 10’s digit is even (9 + 7 = 16), So, 25 write the right side

_____ 25

Step 2: Multiplication of 10’s digit i. e 9 × 7 = 63

And half the sum of the 10’s digit i. e = 8

Step 3: add the 63 and 8 => 63 + 8 = 71

= 71 25

= 7125 (answer)

**3. 35 × 85 = ?**

Step 1: Here, sum of 10’s digit is odd (3 + 8 = 11), So, 75 write the right side

_____ 75

Step 2: Multiplication of 10’s digit i. e 3 × 8 = 24

And half the sum of the 10’s digit i. e = 5

Step 3: add the 24 and 5 => 24 + 5 = 29

= 29 75

= 2975 (answer)

**MULTIPLY OF TWO NUMBERS DIFFERING 2, 4, 6, 8, 10, ………**

**1. [Square of middle number of given numbers] – [1] ^{2} **

**11 × 13 = ?
**Here, the difference between 11 & 13 is = 2 and the middle number is 12

Therefore, 12

^{2}– 1

^{2}=> 144 – 1 = 143 (answer)

**15 × 17 = ?
**Here, the difference between 15 & 17 is = 2 and the middle number is 16

Therefore, 16

^{2}– 1

^{2}=> 256 – 1 = 255 (answer)

**24 × 26 = ?
**Here, the difference between 24 & 26 is = 2 and the middle number is 25

Therefore, 25

^{2}– 1

^{2}=> 625 – 1 = 624 (answer)

**2. [Square of middle number of given numbers] – [2] ^{2} **

**11 × 15 = ?
**Here, the difference between 11 & 15 is = 4 and the middle number is 13

Therefore, 13

^{2}– 2

^{2}=> 169 – 4 = 165 (answer)

**17 × 21 = ?
**Here, the difference between 17 & 21 is = 4 and the middle number is 19

Therefore, 19

^{2}– 2

^{2}=> 361 – 4 = 357 (answer)

**60 × 64 = ?
**Here, the difference between 60 & 64 is = 4 and the middle number is 62

Therefore, 62

^{2}– 2

^{2}=> 3844 – 4 = 3840 (answer)

**3. [Square of middle number of given numbers] – [3] ^{2} **

**11 × 17 = ?
**Here, the difference between 11 & 17 is = 6 and the middle number is 14

Therefore, 14

^{2}– 3

^{2}=> 196 – 9 = 187 (answer)

**13 × 19 = ?
**Here, the difference between 13 & 19 is = 6 and the middle number is 16

Therefore, 16

^{2}– 3

^{2}=> 256 – 9 = 247 (answer)

**4. [Square of a middle number of given numbers] – [4] ^{2} **

**11 × 19 = ?
**Here, the difference between 11 & 19 is = 8 and the middle number is 15

Therefore, 15

^{2}– 4

^{2}=> 225 – 16 = 209 (answer)

**14 × 22 = ?
**Here, the difference between 14 & 22 is = 8 and the middle number is 18

Therefore, 18

^{2}– 4

^{2}=> 324 – 16 = 308 (answer)

**NUMBERS MULTIPLY BY 5, 25, 50, 125, 625:**

**MULTIPLY BY 5:**

**1. 728 × 5 = ?
**728 × 5 = 728 × 5 × 2/2

=728/2 * 10

= 3640 (answer)

**2. 176 × 5 = ?
**176 × 5 = 176 × 5 × 2/2

=176/2 * 10

= 880 (answer)

**MULTIPLY BY 25:**

**1. 728 × 25 = ?
**728 × 25 = 728 × 25 × 4/4

= 728/4 * 100

= 18200 (answer)

**2. 176 × 25 = ?
**176 × 25 = 176 × 25 × 4/4

=176/4 * 100

= 4400 (answer)

**MULTIPLY BY 50:**

**1. 728 × 50 ?**

728 × 50 = 728 × 50 × 2/2

=728/2 * 100

= 36400 (answer)

**2. 176 × 50 = ?
**176 × 50 = 176 × 50 × 2/2

=728/2 * 100

= 36400 (answer)

**MULTIPLY BY 125:**

**1. 728 × 125 = ?
**728 × 125 = 728 × 125 × 8/8

=728/8 * 100

= 91000 (answer)

**2. 176 × 125 = ?
**176 × 125 = 176 × 125 × 8/8

=176/8 * 100

= 22000 (answer)

**NUMBERS DIVISIBLE BY 5, 25, 50, 125, 625:**

**DIVISIBLE BY 5:**

**1.164/5 = ?
**164/5 = 164/5 * 2/2

= 164/10 * 2

= 328/10

= 32. 8 (answer)

**2. 624/5 = ?
**624/5 = 624/5 * 2/2

= 624/10 * 2

= 1248/10

= 124. 8 (answer)

**DIVISIBLE BY 25:**

**1. 164/25 = ?
**164/25 = 164/25 * 4/4

= 164/100 * 4

= 656/100

= 6. 56 (answer)

**2. 624/25 = ?
**624/25 = 624/25 * 4/4

= 624/100 * 4

= 2496/100

= 24. 96 (answer)

**DIVISIBILE BY 50**

**1. 164/50 = ?
**164/50 = 164/50 * 2/2

=164/100 * 2

= 328/100

= 3. 28 (answer)

**2. 624/50 = ?
**624/50 = 624/50 * 2/2

= 624/100 * 2

= 1248/100

= 12. 48 (answer)

**DIVISIBILITY BY 125:**

**1. 164/125 = ?
**164/125 = 164/125 * 8/8

= 164/1000 * 8

= 1312/1000

= 1. 312 (answer)

**2. 624/125 = ?
**624/125 = 624/125 * 8/8

= 624/1000 * 8

= 4992/1000

= 4. 992 (answer)

**Squares:**

**33 ^{2}** = 1 0 8 9

= 1089

**333**^{2} = 11 0 88 9

= 110889

**3333 ^{2}** = 111 0 888 9

= 11108889

**33333 ^{2} **= 1111 0 8888 9

= 1111088889

**99 ^{2} **= 9 8 0 1

= 9801

**999 ^{2} **= 99 8 00 1

= 998001

**9999 ^{2}** = 999 8 000 1

= 99980001

**99999 ^{2} **= 9999 8 0000 1

= 9999800001

**11 ^{2} **= 1 2 1

= 121

**111 ^{2}** = 12 3 21

= 12321

**1111 ^{2}** = 123 4 321

= 1234321

**11111 ^{2}** = 1234 5 4321

= 123454321

111, 222, 333, 444, 555 …………….. are divisible by both 3 and 37

111111, 222222, 333333, 444444 …………….. are divisible by 3, 7, 11, 13 37

**Test of Divisibility: **

**Divisibility by 2:
**A number is divisible by 2 if the unit digit is zero or divisible by 2

Example: 12, 26, 128, 1240 etc.

**Divisibility by 3:
**A number is divisible by 3 if the sum of digits in the number is divisible by 3

Example: 2553

Here, 2 + 5 + 5+ 3 = 15, which is divisible by 3 hence 2553 is divisible by 3

**Divisibility by 4:
**A number is divisible by 4 if its last two digits is divisible by 4

Example: 2652

Here, 52 is divisible by 4, so 2652 is divisible by 4

**Divisibility by 5:
**A number is divisible by 5 if the units digit in number is 0 or 5

Example: 20, 35, 140, 165 etc.

**Divisibility by 6:
**A number is divisible by 6 if the number is even and sum of digits is divisible by 3

Example: 4536

4536 is an even number and also sum of digit 4 + 5 + 3 + 6 = 18 is divisible by 3

**Divisibility by 7:
**To check whether a number is divisible by 7 or not first multiply the units digit of the number by 2 and subtract it from the remaining digits, continue this process. At the end if the result becomes ‘0’ or ‘7’ then the number is divisible by 7

For E.g. : 3066

306 | 6**
**– 12 6 × 2 = 12

**———–**

**29 | 4**

– 8 4 × 2 = 8

**———–**

**2 | 1**

**– 2 1 × 2 = 2**

**———–**

**0**

**———–**

Hence 3066 is divisible by 7

**Divisibility by 8:
**A number is divisible by 8 if last three digit of it is divisible by 8

Example: 47472

Here, 472 is divisible by 8 hence this number 47472 is divisible by 8

**Divisibility by 9:
**A number is divisible by 9 if the sum of its digit is divisible by 9

Example: 108936

Here, 1 + 0 + 8 + 9 + 3 + 6 = 27, which is divisible by 9 and hence 108936 is divisible by 9

**Divisibility by 11:
**A number is divisible by 11 if the difference of sum of digit at odd places and sum of digit at even places is either 0 or divisible by 11

Example: 1331

The sum of digit at odd places is 1 + 3 = 0

And, the sum of digit at even places is 3 + 1 = 0

And, their difference is 4 – 4 = 0

So, 1331 is divisible by 11

**TEST OF DIVISIBILITY BY 13:
**Let us take 2067 … Let us truncate the number

2 0 6 | 7

Add 4 × 7 2 8

2 3 | 4

Add 4 × 4 1 6

3 9, which is divisible by 13

Hence, 2067 is divisible by 13

**TEST OF DIVISIBILITY BY 17:
**Let us take 7752 … Let us truncate the number

7 7 5 | 2

Subtract 5 × 2 1 0

7 6 | 5

Subtract 5 × 5 2 5

5 1, which is divisible by 17

Hence, 7752 is divisible by 17

**TEST OF DIVISIBILITY BY 19:**

Let us take 4864 … Let us truncate the number

4 8 6 | 4

Add 2 × 4 8

——–

4 9 4

4 9 | 4**
**Add 2 × 4 8

——

5 7

57, which is divisible by 19

Hence, 4864 divisible by 19

**10 ^{n} – 1** is always divisible by 11 for all even values of n.

i.e. 99, 9999, 999999 are all divisible by 11 [If there are even digits only]

We know 11

^{2}= 121 Hence 101

^{2}= 10201, 1001

^{2}= 1002001

12

^{2}= 144 Hence 102

^{2}= 10404, 1002

^{2}= 1004004

13

^{2}= 169 Hence 103

^{2}= 10609, 1003

^{2}= 1006009

We know 21

^{2}= 441, 201

^{2}= 40401

31

^{2}= 961, 301

^{2}= 90601

**Condition of Divisibility for Algebraic function:**

**A ^{n} + B^{n}** is exactly divisible by A + B only when n is odd

Example: A

^{3}+ B

^{3}= (A + B)(A

^{2}+ B

^{2}– AB) is divisible by A + B, also A

^{5}+ B

^{5}is divisible by A + B

**A**is never divisible by A – B (whether n- is odd or even)

^{n}+ B^{n}Example: A

^{3}+ B

^{3}= (A + B)(A

^{2}+ B

^{2}– AB) is not divisible by A – B

A

^{7}+ B

^{7}is also not divisible by A – B

**A**is exactly divisible by A- B (whether n- is odd or even)

^{n}– B^{n}Example: A

^{2}– B

^{2}= (A- B) (A+ B) so it is divisible by A- B

A

^{3}– B

^{3}= (A – B)(A

^{2}+ B

^{2}+ AB) so it is divisible by A- B

**Sum of n- natural numbers
**S = 1 + 2 + 3 + 4 + 5 + ……………………. n

S =n (n+1)/2

**Sum of squares of first n- natural numbers
**S = 1

^{2}+ 2

^{2}+ 3

^{2}+ ……………………. N

^{2 }S =n (n+1) (n+2)/2

**Sum of cubes of first n- natural numbers**

S = 1^{3} + 2^{3} + 3^{3} + 4^{3} + ……………….. n^{3
}S= [?? (??+??)/??]??

**Sum of first n- odd natural numbers**

S = 1 + 3 + 5 + 7 + ………………… (2n -1)

S = n^{2}

**Sum of first n- even natural numbers**

S = 2 + 4 + 6 + 8 + ………………… 2n

S = n^{2} + n

v12 + (v12 + (v12+ … ?))

= 4 (answer)

Here, we should find factors for 12 with a difference of 1.

12 = 4 3, the answer is 4

v12 – (v12 – (v12 – … ?)) = 3 (answer = 3)

If the sign is , the answer is 3 (As 12 = 4 3)

v12 (v12 (v12 … n times))= S = [n (n+1)/2]/2

v12 (v12 (v12 … ?))= 12 (answer = 12)

**DIVIDEND = DIVISOR ** **QUOTIENT + REMAINDER**

**DIVISOR** ) DIVIDEND ( **QUOTIENT
** ————-

**Remainder**

**12** ) 170 ( **14**

168

——

2

**170** = 12 * 14 + 2

**ARITHMETIC PROGRESSION (A. P):**

In Mathematics, an Arithmetic Progression or Arithmetic Sequence is a sequence of numbers such that the difference between the consecutive terms is constant

For example, the sequence 5, 8, 11, 14, 17, 20 ……… is an Arithmetic Progression with common difference of 3

**The general form of Arithmetic Progression is**

a, a + d, a + 2d, a + 3d, a + 4d, …………….

First term = a

Common difference = d

No. of terms = n

Any particular term or n – th term = a_{n}

Sum of 1^{st} ‘n’ terms = S_{n}

**a _{n}** = a + (n – 1) * d

**S _{n}** = n/2 * [2a + (n-1) *d]

Or

**S _{n}** = n/2 * ( a + a

_{n})

**GEOMETRIC PROGRESSION:**

In mathematics, a Geometric Progression or Geometric Sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed , non – zero number called the Common ratio

For example, the sequence 2, 6, 18, 54, …………is a Geometric Progression with Common ratio 3

**The general form of a Geometric Progression is,
**a, ar, ar

^{2}, ar

^{3}, ar

^{4}, …………………..

First term = a

Common ration = r

Any particular term or n – th term = a

_{n}

Sum of first n – terms = S

_{n}

**a**= a * rn?¹

_{n}**= a * ( rn?¹ ) /rn?¹ r > 1**

*S*_{n}**S**= a * (1- rn)/1-r r < 1

_{n}**Ramanujan’s Number (1729):**

It is a very interesting number, it is the smallest number expressible as the sum of two cubes in two different ways

1729 = 1^{3} + 12^{3} = 9^{3} + 10^{3}

**Practice Problems**

1. If the expressions x + 809436 × 809438 be a perfect square, then the value of ‘x’ is?

2. The first term of a Geometric Progression (G.P) is 1. The sum of the third and fifth terms is 90. Find the common ratio of the G.P

3. A number when divided by 56, the remainder is 29. If the number is divided by 8, then the remainder is?

4. The sum of two numbers is equal to 200 and their difference is 25. The ratio of the two numbers is?

5. If then the value of a^{2} – 331a is …………..

6. If 1. 5a = 0. 04b, then is equal to ?

7. The number obtained by interchanging the two digits of a two-digit number is lesser than the original number by 54. If the sum of the two digits of the number is 12, then what is the original number?

Solutions for Practice Problems will be available on Fdaytalk Quantitative Aptitude Book, Download Here

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