OPERATION ON NUMBERS
1. Prime Numbers and Composite Numbers:
- Prime Numbers: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, 11, and 13.
- Composite Numbers: A composite number is a natural number greater than 1 that is not prime, meaning it has divisors other than 1 and itself. Examples include 4, 6, 8, 9, and 10.
Example: Determine if 29 is prime or composite.
Solution: 29 is only divisible by 1 and 29 itself, making it a prime number.
2. Factors and Divisors:
- Factors: Factors of a number are the integers that divide the number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
- Divisors: Divisors are the numbers that divide a given number evenly. They are also called factors.
Example:Find the factors of 24.
Solution:The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
3. Prime Factorization:
- Prime Factorization involves expressing a number as a product of its prime factors.
- Example: Prime factorization of 60: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
Example:Find the prime factorization of 84.
Solution: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.
4. Divisibility Rules:
- Divisibility rules help determine if one number is divisible by another without performing actual division.
- Example: Divisibility rule for 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
Example: Is 4752 divisible by 8?
Solution: Since the last three digits (752) are divisible by 8, 4752 is divisible by 8.
5. LCM (Least Common Multiple):
- The LCM of two or more numbers is the smallest multiple that is common to all of them.
- Example: Find the LCM of 12 and 18.
Solution: Prime factorization of 12: 2² × 3, Prime factorization of 18: 2 × 3². LCM = 2² × 3² = 36.
6. GCD (Greatest Common Divisor):
- The GCD of two or more numbers is the largest positive integer that divides all of them.
- Example: Find the GCD of 30 and 45.
Solution:Prime factorization of 30: 2 × 3 × 5, Prime factorization of 45: 3² × 5. GCD = 3 × 5 = 15.
7. Exponents and Powers:
- Exponents indicate how many times a number is multiplied by itself.
- Example: Calculate 5³.
Solution: 5³ = 5 × 5 × 5 = 125.
8. Perfect Squares and Cubes:
- Perfect square is the square of an integer, and a perfect cube is the cube of an integer.
- Example: Determine if 64 is a perfect square.
Solution: 64 is a perfect square since it can be expressed as 8².
9. Triangular Numbers:
- A triangular number is the sum of the first n natural numbers.
- Example: Find the 5th triangular number.
Solution:1 + 2 + 3 + 4 + 5 = 15.
10. Factorial:
- Factorial of a non-negative integer n (denoted as n!) is the product of all positive integers from 1 to n.
- Example: Calculate 4! (4 factorial).
Solution: 4! = 4 × 3 × 2 × 1 = 24.
11. Remainder Theorem:
- The remainder when a number is divided by another number gives insight into the relationship between the two numbers.
- Example: Find the remainder when 257 is divided by 8.
Solution: 257 ÷ 8 = 32 with a remainder of 1.
12. Summation of Series:
- Calculating the sum of a series of numbers involves using formulas to simplify the process.
- Example: Find the sum of the first 10 natural numbers.
Solution: Sum = 1 + 2 + 3 + ... + 10 = 55.
13. Unit Digit:
- The unit digit of a number plays a crucial role in determining the properties of numbers, including divisibility and remainders.
- Example: Find the unit digit of 7⁴.
Solution:The unit digits of the powers of 7 follow a pattern: 7, 9, 3, 1. Therefore, the unit digit of 7⁴ is 1
DIVISIBILITY TESTS
1. Divisibility by 2:
A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
Example: 548 is divisible by 2 because the last digit is 8 (even).
2. Divisibility by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example: 714 is divisible by 3 because 7 + 1 + 4 = 12, which is divisible by 3.
3. Divisibility by 4:
A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
Example: 548 is divisible by 4 because 48 is divisible by 4.
4. Divisibility by 5:
A number is divisible by 5 if its last digit is 0 or 5.
Example: 235 is not divisible by 5 because the last digit is 5 (odd).
5. Divisibility by 6:
A number is divisible by 6 if it is divisible by both 2 and 3.
Example:732 is divisible by 6 because it is divisible by 2 and the sum of its digits is divisible by 3.
6. Divisibility by 7:
There is no simple rule for divisibility by 7, but for larger numbers, it's often easier to perform the division directly.
Example: 175 is not easily determined to be divisible by 7 using a simple rule.
7. Divisibility by 8:
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Example:4568 is divisible by 8 because 568 is divisible by 8.
8. Divisibility by 9:
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example:891 is divisible by 9 because 8 + 9 + 1 = 18, which is divisible by 9.
9. Divisibility by 10:
A number is divisible by 10 if it ends with a 0.
Example:430 is divisible by 10 because it ends with 0.
10. Divisibility by 11:
A number is divisible by 11 if the difference between the sum of the digits in even and odd positions is divisible by 11.
Example: 3742 is divisible by 11 because (3 + 4) - (7 + 2) = -2, which is divisible by 11.
These divisibility tests are useful shortcuts for determining if a number is divisible by another number without performing the actual division..
SQUARE OF NUMBERS 1 TO 50
1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
10^2 = 100
11^2 = 121
12^2 = 144
13^2 = 169
14^2 = 196
15^2 = 225
16^2 = 256
17^2 = 289
18^2 = 324
19^2 = 361
20^2 = 400
21^2 = 441
22^2 = 484
23^2 = 529
24^2 = 576
25^2 = 625
26^2 = 676
27^2 = 729
28^2 = 784
29^2 = 841
30^2 = 900
31^2 = 961
32^2 = 1024
33^2 = 1089
34^2 = 1156
35^2 = 1225
36^2 = 1296
37^2 = 1369
38^2 = 1444
39^2 = 1521
40^2 = 1600
41^2 = 1681
42^2 = 1764
43^2 = 1849
44^2 = 1936
45^2 = 2025
46^2 = 2116
47^2 = 2209
48^2 = 2304
49^2 = 2401
50^2 = 2500
CUBE OF NUMBERS 1 TO 30
1^3 = 1
2^3 = 8
3^3 = 27
4^3 = 64
5^3 = 125
6^3 = 216
7^3 = 343
8^3 = 512
9^3 = 729
10^3 = 1000
11^3 = 1331
12^3 = 1728
13^3 = 2197
14^3 = 2744
15^3 = 3375
16^3 = 4096
17^3 = 4913
18^3 = 5832
19^3 = 6859
20^3 = 8000
21^3 = 9261
22^3 = 10648
23^3 = 12167
24^3 = 13824
25^3 = 15625
26^3 = 17576
27^3 = 19683
28^3 = 21952
29^3 = 24389
30^3 = 27000
PRIME NUMBERS 1 TO 1000
2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97, 101, 103, 107, 109, 113,
127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
179, 181, 191, 193, 197, 199, 211, 223, 227, 229,
233, 239, 241, 251, 257, 263, 269, 271, 277, 281,
283, 293, 307, 311, 313, 317, 331, 337, 347, 349,
353, 359, 367, 373, 379, 383, 389, 397, 401, 409,
419, 421, 431, 433, 439, 443, 449, 457, 461, 463,
467, 479, 487, 491, 499, 503, 509, 521, 523, 541,
547, 557, 563, 569, 571, 577, 587, 593, 599, 601,
607, 613, 617, 619, 631, 641, 643, 647, 653, 659,
661, 673, 677, 683, 691, 701, 709, 719, 727, 733,
739, 743, 751, 757, 761, 769, 773, 787, 797, 809,
811, 821, 823, 827, 829, 839, 853, 857, 859, 863,
877, 881, 883, 887, 907, 911, 919, 929, 937, 941,
947, 953, 967, 971, 977, 983, 991, 997
TRIANGILAR NUMBERS
1, 3, 6, 10, 15, 21, 28, 36, 45, 55,
66, 78, 91, 105, 120, 136, 153, 171, 190, 210,
231, 253, 276, 300, 325, 351, 378, 406, 435, 465,
496, 528, 561, 595, 630, 666, 703, 741, 780, 820,
861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275,
1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1830,
1891, 1953, 2016, 2080, 2145, 2211, 2278, 2346, 2415, 2485,
2556, 2628, 2701, 2775, 2850, 2926, 3003, 3081, 3160, 3240,
3321, 3403, 3486, 3570, 3655, 3741, 3828, 3916, 4005, 4095,
4186, 4278, 4371, 4465, 4560, 4656, 4753, 4851, 4950
FACTORIAL OF NUMBERS
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
9! = 362880
10! = 3628800
Smallest and Largest Numbers Using the Digits 0 to 9:
- Smallest single-digit number: 0
- Largest single-digit number: 9
- Smallest two-digit number: 10
- Largest two-digit number: 99
- Smallest three-digit number: 100
- Largest three-digit number: 999
- Smallest four-digit number: 1000
- Largest four-digit number: 9999
- Smallest five-digit number: 10000
- Largest five-digit number: 99999
- Smallest six-digit number: 100000
- Largest six-digit number: 999999
- Smallest seven-digit number: 1000000
- Largest seven-digit number: 9999999
- Smallest eight-digit number: 10000000
- Largest eight-digit number: 99999999
- Smallest nine-digit number: 100000000
- Largest nine-digit number: 999999999
- Smallest ten-digit number: 1000000000
- Largest ten-digit number: 9999999999
FACTORS OF NUMBERS 1 TO 100
1: 1
2: 1, 2
3: 1, 3
4: 1, 2, 4
5: 1, 5
6: 1, 2, 3, 6
7: 1, 7
8: 1, 2, 4, 8
9: 1, 3, 9
10: 1, 2, 5, 10
11: 1, 11
12: 1, 2, 3, 4, 6, 12
13: 1, 13
14: 1, 2, 7, 14
15: 1, 3, 5, 15
16: 1, 2, 4, 8, 16
17: 1, 17
18: 1, 2, 3, 6, 9, 18
19: 1, 19
20: 1, 2, 4, 5, 10, 20
21: 1, 3, 7, 21
22: 1, 2, 11, 22
23: 1, 23
24: 1, 2, 3, 4, 6, 8, 12, 24
25: 1, 5, 25
26: 1, 2, 13, 26
27: 1, 3, 9, 27
28: 1, 2, 4, 7, 14, 28
29: 1, 29
30: 1, 2, 3, 5, 6, 10, 15, 30
31: 1, 31
32: 1, 2, 4, 8, 16, 32
33: 1, 3, 11, 33
34: 1, 2, 17, 34
35: 1, 5, 7, 35
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
37: 1, 37
38: 1, 2, 19, 38
39: 1, 3, 13, 39
40: 1, 2, 4, 5, 8, 10, 20, 40
41: 1, 41
42: 1, 2, 3, 6, 7, 14, 21, 42
43: 1, 43
44: 1, 2, 4, 11, 22, 44
45: 1, 3, 5, 9, 15, 45
46: 1, 2, 23, 46
47: 1, 47
48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
49: 1, 7, 49
50: 1, 2, 5, 10, 25, 50
51: 1, 3, 17, 51
52: 1, 2, 4, 13, 26, 52
53: 1, 53
54: 1, 2, 3, 6, 9, 18, 27, 54
55: 1, 5, 11, 55
56: 1, 2, 4, 7, 8, 14, 28, 56
57: 1, 3, 19, 57
58: 1, 2, 29, 58
59: 1, 59
60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
61: 1, 61
62: 1, 2, 31, 62
63: 1, 3, 7, 9, 21, 63
64: 1, 2, 4, 8, 16, 32, 64
65: 1, 5, 13, 65
66: 1, 2, 3, 6, 11, 22, 33, 66
67: 1, 67
68: 1, 2, 4, 17, 34, 68
69: 1, 3, 23, 69
70: 1, 2, 5, 7, 10, 14, 35, 70
71: 1, 71
72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
73: 1, 73
74: 1, 2, 37, 74
75: 1, 3, 5, 15, 25, 75
76: 1, 2, 4, 19, 38, 76
77: 1, 7, 11, 77
78: 1, 2, 3, 6, 13, 26, 39, 78
79: 1, 79
80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
81: 1, 3, 9, 27, 81
82: 1, 2, 41, 82
83: 1, 83
84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
85: 1, 5, 17, 85
86: 1, 2, 43, 86
87: 1, 3, 29, 87
88: 1, 2, 4, 8, 11, 22, 44, 88
89: 1, 89
90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
91: 1, 7, 13, 91
92: 1, 2, 4, 23, 46, 92
93: 1, 3, 31, 93
94: 1, 2, 47, 94
95: 1, 5, 19, 95
96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
97: 1, 97
98: 1, 2, 7, 14, 49, 98
99: 1, 3, 9, 11, 33, 99
100: 1, 2, 4, 5, 10, 20, 25, 50, 100
MULTIPLES OF NUMBERS 1 TO 100
1: 1, 2, 3, 4, 5
2: 2, 4, 6, 8, 10
3: 3, 6, 9, 12, 15
4: 4, 8, 12, 16, 20
5: 5, 10, 15, 20, 25
6: 6, 12, 18, 24, 30
7: 7, 14, 21, 28, 35
8: 8, 16, 24, 32, 40
9: 9, 18, 27, 36, 45
10: 10, 20, 30, 40, 50
11: 11, 22, 33, 44, 55
12: 12, 24, 36, 48, 60
13: 13, 26, 39, 52, 65
14: 14, 28, 42, 56, 70
15: 15, 30, 45, 60, 75
16: 16, 32, 48, 64, 80
17: 17, 34, 51, 68, 85
18: 18, 36, 54, 72, 90
19: 19, 38, 57, 76, 95
20: 20, 40, 60, 80, 100
21: 21, 42, 63, 84, 105
22: 22, 44, 66, 88, 110
23: 23, 46, 69, 92, 115
24: 24, 48, 72, 96, 120
25: 25, 50, 75, 100, 125
26: 26, 52, 78, 104, 130
27: 27, 54, 81, 108, 135
28: 28, 56, 84, 112, 140
29: 29, 58, 87, 116, 145
30: 30, 60, 90, 120, 150
31: 31, 62, 93, 124, 155
32: 32, 64, 96, 128, 160
33: 33, 66, 99, 132, 165
34: 34, 68, 102, 136, 170
35: 35, 70, 105, 140, 175
36: 36, 72, 108, 144, 180
37: 37, 74, 111, 148, 185
38: 38, 76, 114, 152, 190
39: 39, 78, 117, 156, 195
40: 40, 80, 120, 160, 200
41: 41, 82, 123, 164, 205
42: 42, 84, 126, 168, 210
43: 43, 86, 129, 172, 215
44: 44, 88, 132, 176, 220
45: 45, 90, 135, 180, 225
46: 46, 92, 138, 184, 230
47: 47, 94, 141, 188, 235
48: 48, 96, 144, 192, 240
49: 49, 98, 147, 196, 245
50: 50, 100, 150, 200, 250
51: 51, 102, 153, 204, 255
52: 52, 104, 156, 208, 260
53: 53, 106, 159, 212, 265
54: 54, 108, 162, 216, 270
55: 55, 110, 165, 220, 275
56: 56, 112, 168, 224, 280
57: 57, 114, 171, 228, 285
58: 58, 116, 174, 232, 290
59: 59, 118, 177, 236, 295
60: 60, 120, 180, 240, 300
61: 61, 122, 183, 244, 305
62: 62, 124, 186, 248, 310
63: 63, 126, 189, 252, 315
64: 64, 128, 192, 256, 320
65: 65, 130, 195, 260, 325
66: 66, 132, 198, 264, 330
67: 67, 134, 201, 268, 335
68: 68, 136, 204, 272, 340
69: 69, 138, 207, 276, 345
70: 70, 140, 210, 280, 350
71: 71, 142, 213, 284, 355
72: 72, 144, 216, 288,360
73: 73, 146, 219, 292, 365
74: 74, 148, 222, 296, 370
75: 75, 150, 225, 300, 375
76: 76, 152, 228, 304, 380
77: 77, 154, 231, 308, 385
78: 78, 156, 234, 312, 390
79: 79, 158, 237, 316, 395
80: 80, 160, 240, 320, 400
81: 81, 162, 243, 324, 405
82: 82, 164, 246, 328, 410
83: 83, 166, 249, 332, 415
84: 84, 168, 252, 336, 420
85: 85, 170, 255, 340, 425
86: 86, 172, 258, 344, 430
87: 87, 174, 261, 348, 435
88: 88, 176, 264, 352, 440
89: 89, 178, 267, 356, 445
90: 90, 180, 270, 360, 450
91: 91, 182, 273, 364, 455
92: 92, 184, 276, 368, 460
93: 93, 186, 279, 372, 465
94: 94, 188, 282, 376, 470
95: 95, 190, 285, 380, 475
96: 96, 192, 288, 384, 480
97: 97, 194, 291, 388, 485
98: 98, 196, 294, 392, 490
99: 99, 198, 297, 396, 495
100: 100, 200, 300, 400, 500
These concepts and examples cover a wide range of number-related topics that are commonly encountered in bank exam questions.
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