Quantitative Aptitude - Number Systems

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Quantitative Aptitude

• Number Systems
• By Fdaytalk

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Fdaytalk.com 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 ?? where X Y are
integers and Y ? 0 are known as
??
rational numbers
Example ??, ??, -?? etc.
?? ?? ??
Irrational numbers Those numbers which when
expressed in decimal form are neither terminating
nor repeating decimals, are known as irrational
numbers. Examples v??, v??, v??, ??
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 which 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 1 to 11 20 format)
11 1 11 12 1 12 13 1 13
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
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11 13 143 12 13 156 13 13 169 13 13 169
11 14 154 12 14 168 13 14 182 13 14 182
11 15 165 12 15 180 13 15 195 13 15 195
11 16 176 12 16 192 13 16 208 13 16 208
11 17 187 12 17 204 13 17 221 13 17 221
11 18 198 12 18 216 13 18 234 13 18 234
11 19 209 12 19 228 13 19 247 13 19 247
11 20 220 12 20 240 13 20 260 13 20 260
14 1 14 15 1 15 15 1 15 16 1 16
14 2 28 15 2 30 15 2 30 16 2 32
14 3 42 15 3 45 15 3 45 16 3 48
14 4 56 15 4 60 15 4 60 16 4 64
14 5 70 15 5 75 15 5 75 16 5 80
14 6 84 15 6 90 15 6 90 16 6 96
14 7 98 15 7 105 15 7 105 16 7 112
14 8 112 15 8 120 15 8 120 16 8 128
14 9 126 15 9 135 15 9 135 16 9 144
14 10 140 15 10 150 15 10 150 16 10 160
14 11 154 15 11 165 15 11 165 16 11 176
14 12 168 15 12 180 15 12 180 16 12 192
14 13 182 15 13 195 15 13 195 16 13 208
14 14 196 15 14 210 15 14 210 16 14 224
14 15 210 15 15 225 15 15 225 16 15 240
14 16 224 15 16 240 15 16 240 16 16 256
14 17 238 15 17 255 15 17 255 16 17 272
14 18 252 15 18 270 15 18 270 16 18 288
14 19 266 15 19 285 15 19 285 16 19 304
14 20 280 15 20 300 15 20 300 16 20 320
17 1 17 18 1 18 18 1 18 19 1 19
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17 2 34 17 2 34 18 2 36 19 2 38 19 2 38
17 3 51 17 3 51 18 3 54 19 3 57 19 3 57
17 4 68 17 4 68 18 4 72 19 4 76 19 4 76
17 5 85 17 5 85 18 5 90 19 5 95 19 5 95
17 6 102 17 6 102 18 6 108 19 6 114 19 6 114
17 7 119 17 7 119 18 7 126 19 7 133 19 7 133
17 8 136 17 8 136 18 8 144 19 8 152 19 8 152
17 9 153 17 9 153 18 9 162 19 9 171 19 9 171
17 10 170 17 10 170 18 10 180 19 10 190 19 10 190
17 11 187 17 11 187 18 11 198 19 11 209 19 11 209
17 12 204 17 12 204 18 12 216 19 12 228 19 12 228
17 13 221 17 13 221 18 13 234 19 13 247 19 13 247
17 14 238 17 14 238 18 14 252 19 14 266 19 14 266
17 15 255 17 15 255 18 15 270 19 15 285 19 15 285
17 16 272 17 16 272 18 16 288 19 16 304 19 16 304
17 17 289 17 17 289 18 17 306 19 17 323 19 17 323
17 18 306 17 18 306 18 18 324 19 18 342 19 18 342
17 19 323 17 19 323 18 19 342 19 19 361 19 19 361
17 20 340 17 20 340 18 20 360 19 20 380 19 20 380
Learn the Square of a numbers from 1 to 99 Learn the Square of a numbers from 1 to 99 Learn the Square of a numbers from 1 to 99 Learn the Square of a numbers from 1 to 99 Learn the Square of a numbers from 1 to 99
012 01 512 2601, 512 2601, 112 121 612 3721
022 04 522 2704, 522 2704, 122 144 622 3844
032 09 532 2809, 532 2809, 132 169 632 3969
042 16 542 2916, 542 2916, 142 196 642 4096
052 25 552 3025, 552 3025, 152 225 652 4225
062 36 562 3136, 562 3136, 162 256 662 4356
072 49 572 3249, 572 3249, 172 289 672 4489
082 64 582 3364, 582 3364, 182 324 682 4624
092 81 592 3481, 592 3481, 192 361 692 4761
212 441 712 5041, 712 5041, 312 961 812 6561
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222 484 722 5184, 322 1024 822 6724
232 529 732 5329, 332 1089 832 6889
242 576 742 5476, 342 1156 842 7056
252 625 752 5625, 352 1225 852 7225
262 676 762 5776, 362 1296 862 7396
272 729 772 5929, 372 1369 872 7569
282 784 782 6084, 382 1444 882 7744
292 841 792 6241, 392 1521 892 7921
412 1681 912 8281 462 2116 962 9216
422 1764 922 8464 472 2209 972 9409
432 1849 932 8649 482 2304 982 9604
442 1936 942 8836 492 2401 992 9801
452 2025 952 9025
Learn the Cube of a numbers from 1 to 30 Learn the Cube of a numbers from 1 to 30 Learn the Cube of a numbers from 1 to 30 Learn the Cube of a numbers from 1 to 30
13 1 113 1331 213 9261 213 9261
23 8 123 1728 223 10648 223 10648
33 27 133 2197 233 12167 233 12167
43 64 143 2744 243 13824 243 13824
53 125 153 3375 253 15625 253 15625
63 216 163 4096 263 17576 263 17576
73 343 173 4913 273 19683 273 19683
83 512 183 5832 283 21952 283 21952
93 729 193 6859 293 24389 293 24389
103 1000 203 8000 303 27000 303 27000
Find out square when unit digit is 5 252 ? 252
625
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Fdaytalk.com (23) (52) 6 25 625
(answer) 352 ? 352 1225 (34) (52) 12 25
1225 (answer) 452 ? 452 2025 (45) (52)
20 25 2025 (answer) 852 ? 852 7225
(89) (52) 72 25 7225 (answer) 952 ? 952
9025 (910) (52) 90 25 9025 (answer) Find
out a square root of a number v8464 ? v8464
92 8464 ? 8100 gt 902 922 Therefore, v8464
92 v???????? ? v9409 97 9409 ? 9025 (we know
that, 952 9025) gt 952 97 Therefore, v9409
97 (answer)
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Fdaytalk.com Find out a square of a number
(106)2 ? (106)2 11236
Step 1 Step 2 Step 3
(06)2 36 1 2 06 12 12 1
Final answer 1 12 36 11236 Therefore, (106)2
11236 (answer) (113)2 ? (113)2 12769
Step 1 Step 2
(13)2 69 (here, 1 is parity or excess) 1 2 13
26
26 1 (parity is added here) 27 12 1
1 27 69 12769 12769 (answer)
Step 3 Final answer
Therefore, (113)2 (209)2 ? (209)2 43681 Step
1
(09)2 81
Step 2 Step 3
2 2 09 36 22 4
4 36 81 43681
Final answer
43681 (answer)
Therefore, (209)2 (216)2 ? (216)2 46656
Step 1 Step 2
(16)2 56 (here 2 is parity or excess) 2 2
16 64 64 2 (parity 2 is added here) 66 (
22 ) 4
Step 3
4 66 56 46656 46656 (answer)
Final answer
Therefore, (216)2
To find out a cube root of a number
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3v1728 ? 3v1728 12 Step 1 in 1728, 8
replaces by 2 (83 512) Therefore, unit place
digit is 2 Step 2 in 1728, ignore 728 (last 3
digits) We now have 1 Here, 1 ? 13 (1 ? 23
) Therefore, tenth place digit is 1
Therefore, 3v1728 12 (answer) ??v??????????
? 3v19683 27 Step 1 in 19683, 3 replaces by 7
(33 27) Therefore, unit place digit is
7 Step 2 in 19683, ignore 683 (last 3 digits)
We now have 19 Here, 19 ? 23 (19 ? 33
) Therefore, tenth place digit is 2
Therefore, 3v19683 27 (answer) TO FIND OUT A
CUBE OF A NUMBER TYPE 1 Number starts with 1
(from left) Example 12, 13, 14, 15,
. (12)3 ? 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
22 4
23 8
1 1
2 2
Double the middle numbers (2 4 only) and add
to them in the same position
Step 3
1 22 42 8
1 4 8 8
1 2 4 8 4 8 --------------------------------------
----- 1 7 (1)2 8
add here
here, parity or excess 1
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added to next left column 1 7 So, (12)3 1728
answer) (13)3 ?
2
8
Step 1 write 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
32 9
33 27
1 1
3 3
Double the middle numbers (3 9 only) and add
to them in the same position
Step 3
1 32 92 27
1 6 18 27
1 3 9 27 6 18 ------------------------------------
------- 2 (1)1 (2)9 (2)7
add here
here, parity or excess 2, 2
1 shown in the brackets added to next left
columns respectively
9 7
2 1 So, (13)3 2197 answer) (15)3 ?
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
52 25
53 125
1 1
5 5
Double the middle numbers (5 25 only) and add
them in the same position
Step 3
1 52 252 125
1 10 50 125
25 125 50
1 5
add here
10
--------------------------------------------------
- 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)
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Fdaytalk.com TYPE 2 Number ends with 1 (from
right)
Example 21, 31, 41, 51 . (21)3 ? Step
1 write given number as it as with some space
2 1 Step 2 Square and Cube the 10th place digit
(2) of a
given number (21) and
write left side to 2 1 23
22 4
2 2
1 1
8
Double the middle numbers (4 2 only) and add
to them in the same position
Step 3
8 8 8
42 8 4
1 1 1
22 4 2 4
add here
8
------------------------------------------- 9 (1)2
6 1 here, parity or excess 1 shown in the
bracket added to next left columns respectively
6 1
9 2 So, (21)3 9261 answer) (41)3 ?
Step 1 write given number as it as with some
space 4 1 Step 2 Square and Cube the 10th place
digit (4) of a
given number (41) and
1
write left side to 4 43 64
42 16
4 4
1 1
Double the middle numbers (16 4 only) and add
them in the same position
Step 3
64 162 42 1
64 32 8 1
16 4 1 32 8
64
add here
------------------------------------------- 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)
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Fdaytalk.com TYPE 3 If both the numbers
same Example 22, 33, 44, 55 .. (22)3 ?
Here, common number is 2 So, write cube of 2 (i.
e 23 8) in four places with spaces 8 8 8 8 Doubl
e the middle numbers (8 8) add them to the same
position
Step 1
Step 2
8 82 16 82 16 8
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
(33)3 ? Step 1
Here, common number is 3 So, write cube of 3 (i.
e 33 27) in four places with some
spaces 27 27 27 27 Double the middle numbers (27
27) add them to the same position
Step 2
272 54
272 27 54 27 27 54
27

27 54
27
add here
--------------------------------------------- 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,
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Fdaytalk.com (23)3 ? Step 1 write the cube of
both the numbers 2 3 (i. e 23 8 33 27)
with the some space 8 27 Step 2 in the middle 8
27, write like 22 3 32 2
8 8
22 3 12
32 2 18
27 27
Step 3 Double the middle numbers (12 18) add
them to the same position
8 122 24 182 36 27
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 ? Step 1 write the cube of both the
numbers 3 5 (i. e 33 27 53 125) with the
some space 27 125 Step 2 in the middle 27 125,
write like 32 5 52 3
27 32 5 52 3 125
27 45 75 125
Step 3
Double the middle numbers (45 75) add them to
the same position
27 452 90 752 150 125
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
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Fdaytalk.com 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 10s digit, here 5 and its next
number i.e 5 6
(5 6) 30
(6 4) 24
3024 (answer) 2) 72 78 ? Step 1
Multiplication of unit digits i. e 2 8 Step 2
Multiplication of 10s digit, here 7 and its next
number i. e 7 8
(7 8) 56
(2 8) 16
5616 (answer) 3) 113 117 ? Step 1
Multiplication of unit digits i. e 3 7 Step 2
Multiplication of 10s digits, here 11 and its
next number i. e 11 12
(3 7) 21
(11 12) 132 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
10s 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
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Fdaytalk.com 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
10s 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 12
2 4 24 24
3 6 36 36
4 8 48 48
5 10 (here, 10s place digit is added to 5) 5 10 (here, 10s place digit is added to 5) 5 10 (here, 10s place digit is added to 5) 5 10 (here, 10s place digit is added to 5)
(5 1) 0 60
6 12 72 6 1 7
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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 (8 2) 10 (10 3)
24 (here, 10s place digit is added to 8 4
104 30 (here, 10s place digit is added to 10)
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 Table of 94 60 260 20 6 26
94 To write a table of 12, just write the tables
of 1 2 in two separate columns
9 4 94 94
18 8 188 188
27 12 (here, 10s place digit is added to 27) 27 12 (here, 10s place digit is added to 27) 27 12 (here, 10s place digit is added to 27) 27 12 (here, 10s 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
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Fdaytalk.com 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 (42 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 (82 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 (142 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 10s place digit and its next
number i. e (6 7) 42 42 25 4225 (answer)
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Fdaytalk.com 2) 125 125 ? Step 1 The number
25 comes right side 25 Step 2 Multiplication
of 10s 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 10s
digits is even number 25 is taken as unit digit
(write right side) in the final answer When sum
of 10s digits is odd number, 75 taken as unit
digit (write right side) in the final answer 1)
45 65 ? Step 1 Here, sum of 10s digit is
even (6 4 10), So, 25 write the right side _
25
Step 2 Multiplication of 10s digit i. e 4 6
24 46
And half the sum of the 10s digit i. e ( ) 5
2
Step 3 add the 24 and 5 gt 24 5 29 29 25
2925 (answer) 2) 95 75 ? Step 1 Here, sum
of 10s digit is even (9 7 16), So, 25 write
the right side _ 25
Step 2 Multiplication of 10s digit i. e 9 7
63 97
And half the sum of the 10s digit i. e ( ) 8
2
Step 3 add the 63 and 8 gt 63 8 71 71 25
7125 (answer) 3) 35 85 ? Step 1 Here, sum of
10s digit is odd (3 8 11), So, 75 write the
right side _ 75
Step 2 Multiplication of 10s digit i. e 3 8
24 38
And half the sum of the 10s digit i. e ( ) 5
2
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• Fdaytalk.com
• Step 3 add the 24 and 5 gt 24 5 29
• 29 75
• 2975 (answer)
• MULTIPLY OF TWO NUMBERS DIFFERING 2, 4, 6, 8, 10,
  
• Square of middle number of given numbers 12
• 11 13 ?
• Here, the difference between 11 13 is 2 and
  the middle number is 12 Therefore, 122 12 gt
  144 1 143 (answer)
• 15 17 ?
• Here, the difference between 15 17 is 2 and
  the middle number is 16 Therefore, 162 12 gt
  256 1 255 (answer)
• 24 26 ?
• Here, the difference between 24 26 is 2 and
  the middle number is 25 Therefore, 252 12 gt
  625 1 624 (answer)
• Square of middle number of given numbers 22
• 11 15 ?
• Here, the difference between 11 15 is 4 and
  the middle number is 13 Therefore, 132 22 gt
  169 4 165 (answer)
• 17 21 ?

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Fdaytalk.com Therefore, 142 32 gt 196 9
187 (answer) 13 19 ? Here, the difference
between 13 19 is 6 and the middle number is
16 Therefore, 162 32 gt 256 9 247
(answer) 4) Square of a middle number of given
numbers 42 11 19 ? Here, the difference
between 11 19 is 8 and the middle number is
15 Therefore, 152 42 gt 225 16 209
(answer) 14 22 ? Here, the difference
between 14 22 is 8 and the middle number is
18 Therefore, 182 42 gt 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 10
2
3640 (answer) 2) 176 5 ?
176 5 176 5 2
2
176 10
2
880 (answer) MULTIPLY BY 25 1) 728 25 ?
728 25 728 25 4
4
728 100
4
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Fdaytalk.com 18200 (answer) 2) 176 25 ?
176 25 176 25 4
4
176 100
4
4400 (answer) MULTIPLY BY 50 1) 728 50 ?
728 50 728 50 2
2
728 100
2
36400 (answer) 2) 176 50 ?
176 50 176 50 2
2
728 100
2
36400 (answer) MULTIPLY BY 125 1) 728 125 ?
728 125 728 125 8
8
728 1000
8
91000 (answer) 2) 176 125 ?
176 125 176 125 8
8
176 1000
8
22000 (answer) NUMBERS DIVISIBLE BY 5, 25, 50,
125, 625 DIVISIBLE BY 5
1) 164 ?
5
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164 164 2 5 5 2 164 2 10 328 10
32. 8 (answer)
2) 624 ?
5
624 624 2 5 5 2 624 2 10 1248 10
124. 8 (answer) DIVISIBLE BY 25
1) 164 ?
25
164 164 4 25 25 4 164 4 100
656 100 6. 56 (answer)
2) 624 ?
25
624 624 4 25 25 4 624 4 100
2496 100 24. 96 (answer) DIVISIBILE BY 50
1) 164 ?
50
164 164 2 50 50 2 164 2 100 328 100
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3. 28 (answer)
2) 624 ?
50
624 624 2 50 50 2 624 2 100
1248 100 12. 48 (answer) DIVISIBILITY BY 125
1) 164 ?
125
164 164 8 125 125 8 164 8 1000
1312 1000 1. 312 (answer)
2) 624 ?
125
624 624 8 125 125 8 624 8 1000
4992 1000 4. 992 (answer)
332 1 0 8 9 1089 3332 11 0 88 9
110889 33332 111 0 888 9 11108889 333332
1111 0 8888 9 1111088889 992 9 8 0 1
9801 9992 99 8 00 1
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Fdaytalk.com 998001 99992 999 8 000 1
99980001 999992 9999 8 0000 1
9999800001 112 1 2 1 121 1112 12 3 21
12321 11112 123 4 321 1234321
111112 1234 5 123454321
4321
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
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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 ----------- 29 4 - 8 ----------- 2 1 -
2 ----------- 0 -----------
6 2 12
4 2 8
1 2 2
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
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Fdaytalk.com TEST OF DIVISIBILITY BY 13 Let us
take 2067 Let us truncate the number 2 0 6 7
2 8 2 3 4 1 6 3 9, which is divisible by 13
Add 4 7
Add 4 4
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 10n 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 112 121 Hence 1012 10201,
10012 1002001 122 144 Hence 1022 10404,
10022 1004004 132 169 Hence 1032 10609,
10032 1006009 We know 212 441, 2012 40401
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Fdaytalk.com 312 961, 3012 90601 Condition
of Divisibility for Algebraic function An Bn is
exactly divisible by A B only when n is
odd Example A3 B3 (A B)(A2 B2 AB) is
divisible by A B, also A5 B5 is divisible by
A B An Bn is never divisible by A B
(whether n- is odd or even) Example A3 B3
(A B)(A2 B2 AB) is not divisible by A -
B A7 B7 is also not divisible by A - B An Bn
is exactly divisible by A- B (whether n- is odd
or even) Example A2 B2 (A- B) (A B) so it
is divisible by A- B A3 - B3 (A - B)(A2 B2
AB) so it is divisible by A- B Sum of n- natural
numbers S 1 2 3 4 5 . n S
?? (????) ?? Sum of squares of first n- natural
numbers S 12 22 32 . N2 S ??
(????)(??????) ?? Sum of cubes of first n-
natural numbers S 13 23 33 43 ..
n3 S ?? (????)?? ?? Sum of first n- odd
natural numbers S 1 3 5 7 (2n
-1) S n2 Sum of first n- even natural
numbers S 2 4 6 8 2n S n2
n We should not forget 1 is neither prime nor
composite. The lowest prime number is 2. The one
and only prime number which is even is 2.
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Fdaytalk.com Two consecutive numbers which are
prime are only 2 and 3. The lowest composite
number is 4. There are 4 prime numbers each
between 1 and 10 and between 10 and 20. 4. The
total number of One digit numbers are 9
(excluding zero) Two digit numbers are 90 Three
digit numbers are 900 Four digit numbers are 9000
v12 v12 v12 ? a 4 (answer) Here, we
should find factors for 12 with a difference of
1. 12 4 3, the answer is 4 v12 - v12 - v12
a 3 (answer 3) If the sign is -, the
answer is 3 (As 12 43)
????-?? 12 ????
v12v12v12 n times
v12v12v12 a
12 (answer 12)
DIVIDEND DIVISOR QUOTIENT REMAINDER DIVISOR
) DIVIDEND ( QUOTIENT ------------- Remainder 12
) 170 ( 14 168 ------ 2 170 12 14
2 ARITHMETIC PROGRESSION (A. P)
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Fdaytalk.com 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 an Sum of
1st n terms Sn an a (n 1) ??
Sn ?? 2?? (?? - 1) ??
2
Or
Sn ?? ( ?? ?? )
??
2
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, ar2, ar3, ar4, .. First term a
Common ration r Any particular term or n th
term an Sum of first n terms Sn an a
????-1
??
?? (?? -1)
S
r gt 1
n
??-1
??
?? (1- ?? )
S
r lt 1
n
1-??
Ramanujans Number (1729) It is a very
interesting number, it is the smallest number
expressible as the sum of two cubes in two
different ways
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Fdaytalk.com ?? 5 If ???? ???? then the value
of a2 331a is .. SOLUTION 1 Given that,
??3 11 a 113 Therefore, a2 331a a(a
331) 1331(1331 331) 1331000 (answer)
6 If 1. 5a 0. 04b, then ??-?? is equal to .,
????
SOLUTION Given that, 1. 5a 0.
04b ?? 1.5 150 ?? 0.04 4 By using Componendo
and Dividend ?? - ?? 150 - 4 ?? ?? 150 4
73 (????????????)
77
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 , SOLUTION Suppose required
number 10x y, where x gt y According to
question (10x y) (10y x) 54 x y 6
(1) And, x y 12 (2) By solving,
equations (1) (2), we get x 9 x y 12 gt
9 y 12 y 3 ? Required number 10 9 3
93 (answer)
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