fdaytalk · fdaytalk.com author: j maha laxmaiah mail id: [email protected] 11 × 13 = 143 12...
TRANSCRIPT
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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 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 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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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
012 = 01 512 = 2601, 112 = 121 612 = 3721
022 = 04 522 = 2704, 122 = 144 622 = 3844
032 = 09 532 = 2809, 132 = 169 632 = 3969
042 = 16 542 = 2916, 142 = 196 642 = 4096
052 = 25 552 = 3025, 152 = 225 652 = 4225
062 = 36 562 = 3136, 162 = 256 662 = 4356
072 = 49 572 = 3249, 172 = 289 672 = 4489
082 = 64 582 = 3364, 182 = 324 682 = 4624
092 = 81 592 = 3481, 192 = 361 692 = 4761
212 = 441 712 = 5041, 312 = 961 812 = 6561
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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
13 = 1 113 = 1331 213 = 9261
23 = 8 123 = 1728 223 = 10648
33 = 27 133 = 2197 233 = 12167
43 = 64 143 = 2744 243 = 13824
53 = 125 153 = 3375 253 = 15625
63 = 216 163 = 4096 263 = 17576
73 = 343 173 = 4913 273 = 19683
83 = 512 183 = 5832 283 = 21952
93 = 729 193 = 6859 293 = 24389
103 = 1000 203 = 8000 303 = 27000
Find out square when unit digit is 5
252 = ?
252 = 625
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
= (2×3) (52)
= 6 25
= 625 (answer)
352 = ?
352 = 1225
= (3×4) (52)
= 12 25
= 1225 (answer)
452 = ?
452 = 2025
= (4×5) (52)
= 20 25
= 2025 (answer)
852 = ?
852 = 7225
= (8×9) (52)
= 72 25
= 7225 (answer)
952 = ?
952 = 9025
= (9×10) (52)
= 90 25
= 9025 (answer)
Find out a square root of a number
√8464 = ?
√8464 = 92
8464 ≅ 8100
> 902
= 922
Therefore, √8464 = 92
√𝟗𝟒𝟎𝟗 = ?
√9409 = 97
9409 ≅ 9025 (we know that, 952 = 9025)
> 952
= 97
Therefore, √9409 = 97 (answer)
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
Find out a square of a number
(106)2 = ?
(106)2 = 11236
Step 1: (06)2 = 36
Step 2: 1× 2× 06 = 12
Step 3: 12 = 1
Final answer = 1 12 36
= 11236
Therefore, (106)2 = 11236 (answer)
(113)2 = ?
(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: 12 = 1
Final answer = 1 27 69
= 12769
Therefore, (113)2 = 12769 (answer)
(209)2 = ?
(209)2 = 43681
Step 1: (09)2 = 81
Step 2: 2 × 2 × 09 = 36
Step 3: 22 = 4
Final answer = 4 36 81
= 43681
Therefore, (209)2 = 43681 (answer)
(216)2 = ?
(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: ( 22 ) = 4
Final answer = 4 66 56
= 46656
Therefore, (216)2 = 46656 (answer)
To find out a cube root of a number
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
√17283
= ?
√17283
= 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,
√17283
= 12 (answer)
√𝟏𝟗𝟔𝟖𝟑𝟑
= ?
√196833
= 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,
√196833
= 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
1 2 22 23
= 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’
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
added to next left column]
1 7 2 8
So, (12)3 = 1728 answer)
(13)3 = ?
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
1 3 32 33
= 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)
(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
1 5 52 53
= 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)
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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 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)
(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
write left side to 4 1
43 42 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)
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
TYPE 3: If both the numbers same
Example: 22, 33, 44, 55 …………………..
(22)3 = ?
Step 1: Here, common number is 2
So, write cube of 2 (i. e 23 = 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
(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
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, ………………………
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
(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 22 × 3 32 × 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 = ?
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 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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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]
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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]
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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 10’s place digit and its next number i. e (6 × 7) = 42
= 42 25
= 4225 (answer)
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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 (4+6
2) = 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 (9+7
2) = 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 (3+8
2) = 5
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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, 122 – 12 => 144 – 1 = 143 (answer)
15 × 17 = ?
Here, the difference between 15 & 17 is = 2 and the middle number is 16
Therefore, 162 – 12 => 256 – 1 = 255 (answer)
24 × 26 = ?
Here, the difference between 24 & 26 is = 2 and the middle number is 25
Therefore, 252 – 12 => 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, 132 – 22 => 169 – 4 = 165 (answer)
17 × 21 = ?
Here, the difference between 17 & 21 is = 4 and the middle number is 19
Therefore, 192 – 22 => 361 – 4 = 357 (answer)
60 × 64 = ?
Here, the difference between 60 & 64 is = 4 and the middle number is 62
Therefore, 622 – 22 => 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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
Therefore, 142 – 32 => 196 – 9 = 187 (answer)
13 × 19 = ?
Here, the difference between 13 & 19 is = 6 and the middle number is 16
Therefore, 162 – 32 => 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, 152 – 42 => 225 – 16 = 209 (answer)
14 × 22 = ?
Here, the difference between 14 & 22 is = 8 and the middle number is 18
Therefore, 182 – 42 => 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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
= 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∗ 1000
= 91000 (answer)
2) 176 × 125 = ?
176 × 125 = 176 × 125 × 8
8
= 176
8∗ 1000
= 22000 (answer)
NUMBERS DIVISIBLE BY 5, 25, 50, 125, 625
DIVISIBLE BY 5
1) 164
5= ?
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
= 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)
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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
= 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 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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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
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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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 = 𝒏 (𝒏+𝟏)
𝟐
Sum of squares of first n- natural numbers
S = 12 + 22 + 32 + ……………………. N2
S = 𝒏 (𝒏+𝟏)(𝟐𝒏+𝟏)
𝟔
Sum of cubes of first n- natural numbers
S = 13 + 23 + 33 + 43 + ……………….. n3
S = [𝒏 (𝒏+𝟏)
𝟐]𝟐
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.
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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
√12 + √12 + √12 + ⋯ α = 4 (answer)
Here, we should find factors for 12 with a difference of 1.
12 = 4 × 3, the answer is 4
√12 − √12 − √12 … … α = 3 (answer = 3)
If the sign is −, the answer is 3 (As 12 = 4×3)
√12√12√12 … … n times = 12𝟐𝒏−𝟏
𝟐𝒏
√12√12√12 … … α = 12 (answer = 12)
DIVIDEND = DIVISOR × QUOTIENT + REMAINDER
DIVISOR ) DIVIDEND ( QUOTIENT
-------------
Remainder
12 ) 170 ( 14
168
------
2
170 = 12 × 14 + 2
ARITHMETIC PROGRESSION (A. P)
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
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) × 𝑑
Sn = 𝑛
2 × [2𝑎 + (𝑛 − 1) × 𝑑]
Or
Sn = 𝑛
2 × ( 𝑎 + 𝑎𝑛)
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 × 𝑟𝑛−1
Sn = 𝑎 ×(𝑟𝑛 −1)
𝑟−1 r > 1
Sn = 𝑎 ×(1− 𝑟𝑛)
1−𝑟 r < 1
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
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
1729 = 13 + 123 = 93 + 103
1 If the expressions x + 809436 × 809438 be a perfect square, then the value
of ‘x’ is …………..
SOLUTION:
Given that, x + 809436 × 809438
= x + (809437 – 1) × (809437 +1)
= x + (809437)2 – (1)2
So, x should be 1 (answer)
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
SOLUTION:
Given that, first term of G.P (a) = 1
∴ ar2 + ar4 = 90 [from question]
1 × r2 + 1 × r4 = 90 [ a = 1]
By solving, we get
r = ± 3 (answer)
3 A number when divided by by 56, the remainder is 29. If the number is
divided by 8, then the remainder is ………..
SOLUTION:
When first divisor is exactly divisible by second divisor, then the second remainder
is obtained by dividing the first remainder by the second divisor.
Therefore, on dividing 29 by 8, the remainder = 5 (answer)
4 The sum of two numbers is equal to 200 and their difference is 25. The
ratio of the two numbers is ……
SOLUTION:
Let the numbers be a and b
According to question
a + b = 200 …………. (1)
a – b = 25 ……..……. (2)
By equation (1) and (2)
𝑎 + 𝑏
𝑎 − 𝑏=
200
25
𝑎 + 𝑏
𝑎 − 𝑏=
8
1
Using Componemdo and Dividendo method
𝑎 + 𝑏 + 𝑎 − 𝑏
𝑎 + 𝑏 − 𝑎 + 𝑏=
8 + 1
8 − 1
𝑎
𝑏=
9
7 (𝑎𝑛𝑠𝑤𝑒𝑟)
Fdaytalk.com
Author: J Maha Laxmaiah Mail id: [email protected]
5 If 𝒂𝟏
𝟑 = 𝟏𝟏 then the value of a2 – 331a is …………..
SOLUTION:
Given that, 𝑎1
3 = 11
a = 113
Therefore, a2 – 331a = a(a – 331)
= 1331(1331 – 331)
= 1331000 (answer)
6 If 1. 5a = 0. 04b, then 𝒃−𝒂
𝒃+𝒂 is equal to …….,
SOLUTION:
Given that, 1. 5a = 0. 04b
𝑏
𝑎=
1.5
0.04=
150
4
By using Componendo and Dividend
𝑏 − 𝑎
𝑏 + 𝑎=
150 − 4
150 + 4
= 73
77 (𝑎𝑛𝑠𝑤𝑒𝑟)
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 > 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 => 9 + y = 12
y = 3
∴ Required number = 10 × 9 + 3
= 93 (answer)