#divisibility rules

[matias-2003]

The divisibility rules of 1 to 100

1: Every number is a multiple of 1

2: The number ends in 0, 2, 4, 6 or 8 (an even digit)

3: The sum of the digits is a multiple of 3

4: The 2nd to last digit is even and the last digit is 0, 4 or 8, or the 2nd to last digit is odd and the last digit is 2 or 6

5: The number ends in 0 or 5

6: The number is a multiple of both 2 and 3

7: The subtraction of twice the last digit and the rest of the number is a multiple of 7

8: The 3rd to last digit is even and the last 2 digits are a multiple of 8 or the 3rd to last digit is odd and the last 2 digits are a multiple of 4 but not of 8

9: The sum of the digits is a multiple of 9

10: The number ends in 0

11: The subtraction of the sum of the digits in odd places and the sum of the digits in even places is a multiple of 11

12: The number is a multiple of both 3 and 4

13: The sum of 4 times the last digit and the rest of the number is a multiple of 13

14: The number is a multiple of both 2 and 7

15: The number is a multiple of both 3 and 5

16: The 4th to last digit is even and the last 3 digits are a multiple of 16 or the 4th to last digit is odd and the last 3 digits are a multiple of 8 but not of 16

17: The subtraction of 5 times the last digit and the rest of the number is a multiple of 17

18: The number is a multiple of both 2 and 9

19: The sum of twice the last digit and the rest of the number is a multiple of 19

20: The number ends in 00, 20, 40, 60 or 80

21: The subtraction twice the last digit and the rest of the number is a multiple of 21

22: The number is a multiple of both 2 and 11

23: The sum of 7 times the last digit and the rest of the number is a multiple of 23

24: The number is a multiple of both 3 and 8

25: The number ends in 00, 25, 50 or 75

26: The number is a multiple of both 2 and 13

27: The subtracrion of 8 times the last digit and the rest of the number is a multiple of 27

28: The number is a multiple of both 4 and 7

29: The sum of 3 times the last digit and the rest of the number is a multiple of 29

30: The number is a multiple of both 3 and 10

31: The subtraction of 3 times the last digit and the rest of the number is a multiple of 31

32: The last 5 digits are a multiple of 32

33: The sum of 10 times the last digit and the rest of the number is a multiple of 33

34: The number is a multiple of both 2 and 17

35: The number is a multiple of both 5 and 7

36: The number is a multiple of both 4 and 9

37: The subtraction of 11 times the last digit and the rest of the number is a multiple of 37

38: The number is a multiple of both 2 and 19

39: The sum of 4 times the last digit and the rest of the number is a multiple of 39

40: The 3rd to last digit is even and the last 2 digits are 00, 40 or 80, or the 3rd to last digit is odd and the last 2 digits are 20 or 60

41: The subtraction of 4 times the last digit and the rest of the number is a multiple of 41

42: The number is a multiple of both 2 and 21

43: The sum of 13 times the last digit and the rest of the number is a multiple of 43

44: The number is a multiple of both 4 and 11

45: The number is a multiple of both 5 and 9

46: The number is a multiple of both 2 and 23

47: The subtraction of 14 times the last digit and the rest of the number is a multiple of 47

48: The number is a multiple of both 3 and 16

49: The sum of 5 times the last digit and the rest of the number is a multiple of 49

50: The number ends in 00 or 50

51: The subtraction of 5 times the last digit and the rest of the number is a multiple of 51

52: The number is a multiple of both 4 and 13

53: The sum of 16 times the last digit and the rest of the number is a multiple of 53

54: The number is a multiple of both 2 and 27

55: The number is a multiple of both 5 and 11

56: The number is a multiple of both 7 and 8

57: The subtraction of 17 times the last digit and the rest of the number is a multiple of 57

58: The number is a multiple of both 2 and 29

59: The sum of 6 times the last digit and the rest of the number is a multiple of 59

60: The number is a multiple of both 3 and 20

61: The subtraction of 6 times the last digit and the rest of the number is a multiple of 61

62: The number is a multiple of both 2 and 31

63: The sum of 19 times the last digit and the rest of the number is a multiple of 63

64: The last 6 digits are a multiple of 64

65: The number is a multiple of both 5 and 13

66: The number is a multiple of both 2 and 33

67: The subtraction of 20 times the last digit and the rest of the number is a multiple of 67

68: The number is a multiple of both 4 and 17

69: The sum of 7 times the last digit and the rest of the number is a multiple of 69

70: The number is a multiple of both 7 and 10

71: The subtraction of 7 times the last digit and the rest of the number is a multiple of 71

72: The number is a multiple of both 8 and 9

73: The sum of 22 times the last digit and the rest of the number is a multiple of 73

74: The number is a multiple of both 2 and 37

75: The number is a multiple of both 3 and 25

76: The number is a multiple of both 4 and 19

77: The subtraction of 23 times the last digit and the rest of the number is a multiple of 77

78: The number is a multiple of both 2 and 39

79: The sum of 8 times the last digit and the rest of the number is a multiple of 79

80: The 4th to last digit is even and the last 3 digits are a multiple of 80 or the 4th to last digit is odd and the last 3 digits are a multiple of 40 but not of 80

81: The subtraction of 8 times the last digit and the rest of the number is a multiple of 81

82: The number is a multiple of both 2 and 41

83: The sum of 25 times the last digit and the rest of the number is a multiple of 83

84: The number is a multiple of both 4 and 21

85: The number is a multiple of both 5 and 17

86: The number is a multiple of both 2 and 43

87: The subtraction of 26 times the last digit and the rest of the number is a multiple of 87

88: The number is a multiple of both 8 and 11

89: The sum of 9 times the last digit and the rest of the number is a multiple of 89

90: The number is a multiple of both 9 and 10

91: The subtraction of 9 times the last digit and the rest of the number is a multiple of 91

92: The number is a multiple of both 4 and 23

93: The sum of 28 times the last digit and the rest of the number is a multiple of 93

94: The number is a multiple of both 2 and 47

95: The number is a multiple of both 5 and 19

96: The number is a multiple of both 3 and 32

97: The subtraction of 29 times the last digit and the rest of the number is a multiple of 97

98: The number is a multiple of both 2 and 49

99: The sum of 10 times the last digit and the rest of the number is a multiple of 99

100: The number ends in 00

[matias-2003]

The divisibility rules of every number from 2001 to 2100

2001: The number is a multiple of 3, 23 and 29 at the same time

2002: The number is a multiple of 2, 7, 11 and 13 at the same time

2003: The sum of 601 times the last digit and the rest of the number is a multiple of 2003

2004: The number is a multiple of 3, 4 and 167 at the same time

2005: The number is a multiple of 5 and 401 at the same time

2006: The number is a multiple of 2, 17 and 59 at the same time

2007: The number is a multiple of 9 and 223 at the same time

2008: The number is a multiple of 8 and 251 at the same time

2009: The number is a multiple of 41 and 49 at the same time

2010: The number is a multiple of 3, 10 and 67 at the same time

2011: The difference between 201 times the last digit and the rest of the number is a multiple of 2011

2012: The number is a multiple of 4 and 503 at the same time

2013: The number is a multiple of 3, 11 and 61 at the same time

2014: The number is a multiple of 2, 19 and 53 at the same time

2015: The number is a multiple of 5, 13 and 31 at the same time

2016: The number is a multiple of 7, 9 and 32 at the same time

2017: The difference between 605 times the last digit and the rest of the number is a multiple of 2017

2018: The number is a multiple of 2 and 1009 at the same time

2019: The number is a multiple of 3 and 673 at the same time

2020: The number is a multiple of 20 and 101 at the same time

2021: The number is a multiple of 43 and 47 at the same time

2022: The number is a multiple of 2, 3 and 337 at the same time

2023: The number is a multiple of 7 and 289 at the same time

2024: The number is a multiple of 8, 11 and 23 at the same time

2025: The number is a multiple of 25 and 81 at the same time

2026: The number is a multiple of 2 and 1013 at the same time

2027: The difference between 608 times the last digit and the rest of the number is a multiple of 2027

2028: The number is a multiple of 3, 4 and 169 at the same time

2029: The sum of 203 times the last digit and the rest of the number is a multiple of 2029

2030: The number is a multiple of 7, 10 and 29 at the same time

2031: The number is a multiple of 3 and 677 at the same time

2032: The number is a multiple of 16 and 127 at the same time

2033: The number is a multiple of 19 and 107 at the same time

2034: The number is a multiple of 2, 9 and 113 at the same time

2035: The number is a multiple of 5, 11 and 37 at the same time

2036: The number is a multiple of 4 and 509 at the same time

2037: The number is a multiple of 3, 7 and 97 at the same time

2038: The number is a multiple of 2 and 1019 at the same time

2039: The sum of 204 times the last digit and the rest of the number is a multiple of 2039

2040: The number is a multiple of 3, 17 and 40 at the same time

2041: The number is a multiple of 13 and 157 at the same time

2042: The number is a multiple of 2 and 1021 at the same time

2043: The number is a multiple of 9 and 227 at the same time

2044: The number is a multiple of 4, 7 and 73 at the same time

2045: The number is a multiple of 5 and 409 at the same time

2046: The number is a multiple of 2, 3, 11 and 31 at the same time

2047: The number is a multiple of 23 and 89 at the same time

2048: The last 11 digits are a multiple of 2048

The 10,000,000,000s digit is even and the last 10 digits are a multiple of 2048

The 10,000,000,000s digit is odd and the last 10 digits are 1024 times an odd number

2049: The number is a multiple of 3 and 683 at the same time

2050: The number is a multiple of 41 and 50 at the same time

2051: The number is a multiple of 7 and 293 at the same time

2052: The number is a multiple of 4, 19 and 27 at the same time

2053: The sum of 616 times the last digit and the rest of the number is a multiple of 2053

2054: The number is a multiple of 2, 13 and 79 at the same time

2055: The number is a multiple of 3, 5 and 137 at the same time

2056: The number is a multiple of 8 and 257 at the same time

2057: The number is a multiple of 17 and 121 at the same time

2058: The number is a multiple of 2, 3 and 343 at the same time

2059: The number is a multiple of 29 and 71 at the same time

2060: The number is a multiple of 20 and 103 at the same time

2061: The number is a multiple of 9 and 229 at the same time

2062: The number is a multiple of 2 and 1031 at the same time

2063: The sum of 619 times the last digit and the rest of the number is a multiple of 2063

2064: The number is a multiple of 3, 16 and 43 at the same time

2065: The number is a multiple of 5, 7 and 59 at the same time

2066: The number is a multiple of 2 and 1033 at the same time

2067: The number is a multiple of 3, 13 and 53 at the same time

2068: The number is a multiple of 4, 11 and 47 at the same time

2069: The sum of 207 times the last digit and the rest of the number is a multiple of 2069

2070: The number is a multiple of 9, 10 and 23 at the same time

2071: The number is a multiple of 19 and 109 at the same time

2072: The number is a multiple of 7, 8 and 37 at the same time

2073: The number is a multiple of 3 and 691 at the same time

2074: The number is a multiple of 2, 17 and 61 at the same time

2075: The number is a multiple of 25 and 83 at the same time

2076: The number is a multiple of 3, 4 and 173 at the same time

2077: The number is a multiple of 31 and 67 at the same time

2078: The number is a multiple of 2 and 1039 at the same time

2079: The number is a multiple of 7, 11 and 27 at the same time

2080: The number is a multiple of 13 and 160 at the same time

2081: The difference between 208 times the last digit and the rest of the number is a multiple of 2081

2082: The number is a multiple of 2, 3 and 347 at the same time

2083: The sum of 625 times the last digit and the rest of the number is a multiple of 2083

2084: The number is a multiple of 4 and 521 at the same time

2085: The number is a multiple of 3, 5 and 139 at the same time

2086: The number is a multiple of 2, 7 and 149 at the same time

2087: The difference between 626 times the last digit and the rest of the number is a multiple of 2087

2088: The number is a multiple of 8, 9 and 29 at the same time

2089: The sum of 209 times the last digit and the rest of the number is a multiple of 2089

2090: The number is a multiple of 10, 11 and 19 at the same time

2091: The number is a multiple of 3, 17 and 41 at the same time

2092: The number is a multiple of 4 and 523 at the same time

2093: The number is a multiple of 7, 13 and 23 at the same time

2094: The number is a multiple of 2, 3 and 349 at the same time

2095: The number is a multiple of 5 and 419 at the same time

2096: The number is a multiple of 16 and 131 at the same time

2097: The number is a multiple of 9 and 233 at the same time

2098: The number is a multiple of 2 and 1049 at the same time

2099: The sum of 210 times the last digit and the rest of the number is a multiple of 2099

2100: The number is a multiple of 3, 7 and 100 at the same time

[matias-2003]

The divisibility rules of every number from 1 to 100 with alternate methods for the numbers that end in 1, 3, 7 or 9

1: Every number is a multiple of 1

2: The number ends in 0, 2, 4, 6 or 8

3: The sum of the digits is a multiple of 3

4: The last 2 digits are a multiple of 4

The 10s digit is even and the last digit is 0, 4 or 8

The 10s digit is odd and the last digit is 2 or 6

5: The number ends in 0 or 5

6: The number is a multiple of 2 and 3 at the same time

7: The difference between twice the last digit and the rest of the number is a multiple of 7

8: The last 3 digits are a multiple of 8

The 100s digit is even and the last 2 digits are a multiple of 8

The 100s digit is odd and the last 2 digits are 4 times an odd number

9: The sum of the digits is a multiple of 9

10: The number ends in 0

11: The difference between the last digit and the rest of the number is a multiple of 11

12: The number is a multiple of 3 and 4 at the same time

13: The sum of 4 times the last digit and the rest of the number is a multiple of 13

14: The number is a multiple of 2 and 7 at the same time

15: The number is a multiple of 3 and 5 at the same time

16: The last 4 digits are a multiple of 16

The 1,000s digit is even and the last 3 digits are a multiple of 16

The 1,000s digit is odd and the last 3 digits are 8 times an odd number

17: The difference between 5 times the last digit and the rest of the number is a multiple of 17

18: The number is a multiple of 2 and 9 at the same time

19: The sum of twice the last digit and the rest of the number is a multiple of 19

20: The number ends in 00, 20, 40, 60 or 80

21: The difference between twice the last digit and the rest of the number is a multiple of 21

22: The number is a multiple of 2 and 11 at the same time

23: The sum of 7 times the last digit and the rest of the number is a multiple of 23

24: The number is a multiple of 3 and 8 at the same time

25: The number ends in 00, 25, 50 or 75

26: The number is a multiple of 2 and 13 at the same time

27: The difference between 8 times the last digit and the rest of the number is a multiple of 27

28: The number is a multiple of 4 and 7 at the same time

29: The sum of thrice the last digit and the rest of the number is a multiple of 29

30: The number is a multiple of 3 and 10 at the same time

31: The difference between thrice the last digit and the rest of the number is a multiple of 31

32: The last 5 digits are a multiple of 32

The 10,000s digit is even and the last 4 digits are a multiple of 32

The 10,000s digit is odd and the last 4 digits are 16 times an odd number

33: The sum of 10 times the last digit and the rest of the number is a multiple of 33

34: The number is a multiple of 2 and 17 at the same time

35: The number is a multiple of 5 and 7 at the same time

36: The number is a multiple of 4 and 9 at the same time

37: The difference between 11 times the last digit and the rest of the number is a multiple of 37

38: The number is a multiple of 2 and 19 at the same time

39: The sum of 4 times the last digit and the rest of the number is a multiple of 39

40: The last 3 digits are a multiple of 40

The 100s digit is even and the last 2 digits are 00, 40 or 80

The 100s digit is odd and the last 2 digits are 20 or 60

41: The difference between 4 times the last digit and the rest of the number is a multiple of 41

42: The number is a multiple of 2 and 21 at the same time

43: The sum of 13 times the last digit and the rest of the number is a multiple of 43

44: The number is a multiple of 4 and 11 at the same time

45: The number is a multiple of 5 and 9 at the same time

46: The number is a multiple of 2 and 23 at the same time

47: The difference between 14 times the last digit and the rest of the number is a multiple of 47

48: The number is a multiple of 3 and 16 at the same time

49: The sum of 5 times the last digit and the rest of the number is a multiple of 49

50: The number ends in 00 or 50

51: The difference between 5 times the last digit and the rest of the number is a multiple of 51

52: The number is a multiple of 4 and 13 at the same time

53: The sum of 16 times the last digit and the rest of the number is a multiple of 53

54: The number is a multiple of 2 and 27 at the same time

55: The number is a multiple of 5 and 11 at the same time

56: The number is a multiple of 7 and 8 at the same time

57: The difference between 17 times the last digit and the rest of the number is a multiple of 57

58: The number is a multiple of 2 and 29 at the same time

59: The sum of 6 times the last digit and the rest of the number is a multiple of 59

60: The number is a multiple of 3 and 20 at the same time

61: The difference between 6 times the last digit and the rest of the number is a multiple of 61

62: The number is a multiple of 2 and 31 at the same time

63: The sum of 19 times the last digit and the rest of the number is a multiple of 63

64: The last 6 digits are a multiple of 64

The 100,000s digit is even and the last 5 digits are a multiple of 64

The 100,000s digit is odd and the last 5 digits are 32 times an odd number

65: The number is a multiple of 5 and 13 at the same time

66: The number is a multiple of 2 and 33 at the same time

67: The difference between 20 times the last digit and the rest of the number is a multiple of 67

68: The number is a multiple of 4 and 17 at the same time

69: The sum of 7 times the last digit and the rest of the number is a multiple of 69

70: The number is a multiple of 7 and 10 at the same time

71: The difference between 7 times the last digit and the rest of the number is a multiple of 71

72: The number is a multiple of 8 and 9 at the same time

73: The sum of 22 times the last digit and the rest of the number is a multiple of 73

74: The number is a multiple of 2 and 37 at the same time

75: The number is a multiple of 3 and 25 at the same time

76: The number is a multiple of 4 and 19 at the same time

77: The difference between 23 times the last digit and the rest of the number is a multiple of 77

78: The number is a multiple of 2 and 39 at the same time

79: The sum of 8 times the last digit and the rest of the number is a multiple of 79

80: The last 4 digits are a multiple of 80

The 1,000s digit is even and the last 3 digits are a multiple of 80

The 1,000s digit is odd and the last 3 digits are 40 times an odd number

81: The difference between 8 times the last digit and the rest of the number is a multiple of 81

82: The number is a multiple of 2 and 41 at the same time

83: The sum of 25 times the last digit and the rest of the number is a multiple of 83

84: The number is a multiple of 4 and 21 at the same time

85: The number is a multiple of 5 and 17 at the same time

86: The number is a multiple of 2 and 43 at the same time

87: The difference between 26 times the last digit and the rest of the number is a multiple of 87

88: The number is a multiple of 8 and 11 at the same time

89: The sum of 9 times the last digit and the rest of the number is a multiple of 89

90: The number is a multiple of 9 and 10 at the same time

91: The difference between 9 times the last digit and the rest of the number is a multiple of 91

92: The number is a multiple of 4 and 23 at the same time

93: The sum of 28 times the last digit and the rest of the number is a multiple of 93

94: The number is a multiple of 2 and 47 at the same time

95: The number is a multiple of 5 and 19 at the same time

96: The number is a multiple of 3 and 32 at the same time

97: The difference between 29 times the last digit and the rest of the number is a multiple of 97

98: The number is a multiple of 2 and 49 at the same time

99: The sum of 10 times the last digit and the rest of the number is a multiple of 99

100: The number ends in 00

[matias-2003]

The divisibility rules of the 1st 10 powers of 5

5: The number ends in 0 or 5

25: The number ends in 00, 25, 50 or 75

125: The number ends in 000, 125, 250, 375, 500, 625, 750 or 875

625: The last 4 digits are a multiple of 625

The 1,000s digit is 0 or 5 and the last 3 digits are 000 or 625

The 1,000s digit is 1 or 6 and the last 3 digits are 250 or 875

The 1,000s digit is 2 or 7 and the last 3 digits are 500

The 1,000s digit is 3 or 8 and the last 3 digits are 125 or 750

The 1,000s digit is 4 or 9 and the last 3 digits are 375

3125: The last 5 digits are a multiple of 3125

The 10,000s digit is 0 or 5 and the last 4 digits are a multiple of 3125

The 10,000s digit is 1 or 6 and the last 4 digits are 625 times a number that ends in 4 or 9

The 10,000s digit is 2 or 7 and the last 4 digits are 625 times a number that ends in 3 or 8

The 10,000s digit is 3 or 8 and the last 4 digits are 625 times a number that ends in 2 or 7

The 10,000s digit is 4 or 9 and the last 4 digits are 625 times a number that ends in 1 or 6

15625: The last 6 digits are a multiple of 15625

The 100,000s digit is 0 or 5 and the last 5 digits are a multiple of 15625

The 100,000s digit is 1 or 6 and the last 5 digits are 3125 times a number that ends in 3 or 8

The 100,000s digit is 2 or 7 and the last 5 digits are 3125 times a number that ends in 1 or 6

The 100,000s digit is 3 or 8 and the last 5 digits are 3125 times a number that ends in 4 or 9

The 100,000s digit is 4 or 9 and the last 5 digits are 3125 times a number that ends in 2 or 7

78125: The last 7 digits are a multiple of 78125

The 1,000,000s digit is 0 or 5 and the last 6 digits are a multiple of 78125

The 1,000,000s digit is 1 or 6 and the last 6 digits are 15625 times a number that ends in 1 or 6

The 1,000,000s digit is 2 or 7 and the last 6 digits are 15625 times a number that ends in 2 or 7

The 1,000,000s digit is 3 or 8 and the last 6 digits are 15625 times a number that ends in 3 or 8

The 1,000,000s digit is 4 or 9 and the last 6 digits are 15625 times a number that ends in 4 or 9

390625: The last 8 digits are a multiple of 390625

The 10,000,000s digit is 0 or 5 and the last 7 digits are a multiple of 390625

The 10,000,000s digit is 1 or 6 and the last 7 digits are 78125 times a number that ends in 2 or 7

The 10,000,000s digit is 2 or 7 and the last 7 digits are 78125 times a number that ends in 4 or 9

The 10,000,000s digit is 3 or 8 and the last 7 digits are 78125 times a number that ends in 1 or 6

The 10,000,000s digit is 4 or 9 and the last 7 digits are 78125 times a number that ends in 3 or 8

1953125: The last 9 digits are a multiple of 1953125

The 100,000,000s digit is 0 or 5 and the last 8 digits are a multiple of 1953125

The 100,000,000s digit is 1 or 6 and the last 8 digits are 390625 times a number that ends in 4 or 9

The 100,000,000s digit is 2 or 7 and the last 8 digits are 390625 times a number that ends in 3 or 8

The 100,000,000s digit is 3 or 8 and the last 8 digits are 390625 times a number that ends in 2 or 7

The 100,000,000s digit is 4 or 9 and the last 8 digits are 390625 times a number that ends in 1 or 6

9765625: The last 10 digits are a multiple of 9765625

The 1,000,000,000s digit is 0 or 5 and the last 9 digits are a multiple of 9765625

The 1,000,000,000s digit is 1 or 6 and the last 9 digits are 1953125 times a number that ends in 3 or 8

The 1,000,000,000s digit is 2 or 7 and the last 9 digits are 1953125 times a number that ends in 1 or 6

The 1,000,000,000s digit is 3 or 8 and the last 9 digits are 1953125 times a number that ends in 4 or 9

The 1,000,000,000s digit is 4 or 9 and the last 9 digits are 1953125 times a number that ends in 2 or 7

[matias-2003]

The divisibility rules of every number from 2101 to 2200

2101: The number is a multiple of 11 and 191 at the same time

2102: The number is a multiple of 2 and 1051 at the same time

2103: The number is a multiple of 3 and 701 at the same time

2104: The number is a multiple of 8 and 263 at the same time

2105: The number is a multiple of 5 and 421 at the same time

2106: The number is a multiple of 2, 13 and 81 at the same time

2107: The number is a multiple of 43 and 49 at the same time

2108: The number is a multiple of 4, 17 and 31 at the same time

2109: The number is a multiple of 3, 19 and 37 at the same time

2110: The number is a multiple of 10 and 211 at the same time

2111: The difference between 211 times the last digit and the rest of the number is a multiple of 2111

2112: The number is a multiple of 3, 11 and 64 at the same time

2113: The sum of 634 times the last digit and the rest of the number is a multiple of 2113

2114: The number is a multiple of 2, 7 and 151 at the same time

2115: The number is a multiple of 5, 9 and 47 at the same time

2116: The number is a multiple of 4 and 529 at the same time

2117: The number is a multiple of 29 and 73 at the same time

2118: The number is a multiple of 2, 3 and 353 at the same time

2119: The number is a multiple of 13 and 163 at the same time

2120: The number is a multiple of 40 and 53 at the same time

2121: The number is a multiple of 3, 7 and 101 at the same time

2122: The number is a multiple of 2 and 1061 at the same time

2123: The number is a multiple of 11 and 193 at the same time

2124: The number is a multiple of 4, 9 and 59 at the same time

2125: The number is a multiple of 17 and 125 at the same time

2126: The number is a multiple of 2 and 1063 at the same time

2127: The number is a multiple of 3 and 709 at the same time

2128: The number is a multiple of 7, 16 and 19 at the same time

2129: The sum of 213 times the last digit and the rest of the number is a multiple of 2129

2130: The number is a multiple of 3, 10 and 71 at the same time

2131: The difference between 213 times the last digit and the rest of the number is a multiple of 2131

2132: The number is a multiple of 4, 13 and 41 at the same time

2133: The number is a multiple of 27 and 79 at the same time

2134: The number is a multiple of 2, 11 and 97 at the same time

2135: The number is a multiple of 5, 7 and 61 at the same time

2136: The number is a multiple of 3, 8 and 89 at the same time

2137: The difference between 641 times the last digit and the rest of the number is a multiple of 2137

2138: The number is a multiple of 2 and 1069 at the same time

2139: The number is a multiple of 3, 23 and 31 at the same time

2140: The number is a multiple of 20 and 107 at the same time

2141: The difference between 214 times the last digit and the rest of the number is a multiple of 2141

2142: The number is a multiple of 2, 7, 9 and 17 at the same time

2143: The sum of 643 times the last digit and the rest of the number is a multiple of 2143

2144: The number is a multiple of 32 and 67 at the same time

2145: The number is a multiple of 3, 5, 11 and 13 at the same time

2146: The number is a multiple of 2, 29 and 37 at the same time

2147: The number is a multiple of 19 and 113 at the same time

2148: The number is a multiple of 3, 4 and 179 at the same time

2149: The number is a multiple of 7 and 307 at the same time

2150: The number is a multiple of 43 and 50 at the same time

2151: The number is a multiple of 9 and 239 at the same time

2152: The number is a multiple of 8 and 269 at the same time

2153: The sum of 646 times the last digit and the rest of the number is a multiple of 2153

2154: The number is a multiple of 2, 3 and 353 at the same time

2155: The number is a multiple of 5 and 431 at the same time

2156: The number is a multiple of 4, 11 and 49 at the same time

2157: The number is a multiple of 3 and 719 at the same time

2158: The number is a multiple of 2, 13 and 83 at the same time

2159: The number is a multiple of 17 and 127 at the same time

2160: The number is a multiple of 27 and 80 at the same time

2161: The difference between 216 times the last digit and the rest of the number is a multiple of 2161

2162: The number is a multiple of 2, 23 and 47 at the same time

2163: The number is a multiple of 3, 7 and 103 at the same time

2164: The number is a multiple of 4 and 541 at the same time

2165: The number is a multiple of 5 and 433 at the same time

2166: The number is a multiple of 2, 3 and 361 at the same time

2167: The number is a multiple of 11 and 197 at the same time

2168: The number is a multiple of 8 and 271 at the same time

2169: The number is a multiple of 9 and 241 at the same time

2170: The number is a multiple of 7, 10 and 31 at the same time

2171: The number is a multiple of 13 and 167 at the same time

2172: The number is a multiple of 3, 4 and 181 at the same time

2173: The number is a multiple of 41 and 53 at the same time

2174: The number is a multiple of 2 and 1087 at the same time

2175: The number is a multiple of 3, 25 and 29 at the same time

2176: The number is a multiple of 17 and 128 at the same time

2177: The number is a multiple of 7 and 311 at the same time

2178: The number is a multiple of 2, 9 and 121 at the same time

2179: The sum of 218 times the last digit and the rest of the number is a multiple of 2179

2180: The number is a multiple of 20 and 109 at the same time

2181: The number is a multiple of 3 and 727 at the same time

2182: The number is a multiple of 2 and 1091 at the same time

2183: The number is a multiple of 37 and 59 at the same time

2184: The number is a multiple of 3, 7, 8 and 13 at the same time

2185: The number is a multiple of 5, 19 and 23 at the same time

2186: The number is a multiple of 2 and 1093 at the same time

2187: The difference between 656 times the last digit and the rest of the number is a multiple of 2187

2188: The number is a multiple of 4 and 547 at the same time

2189: The number is a multiple of 11 and 199 at the same time

2190: The number is a multiple of 3, 10 and 73 at the same time

2191: The number is a multiple of 7 and 313 at the same time

2192: The number is a multiple of 16 and 137 at the same time

2193: The number is a multiple of 3, 17 and 43 at the same time

2194: The number is a multiple of 2 and 1097 at the same time

2195: The number is a multiple of 5 and 439 at the same time

2196: The number is a multiple of 4, 9 and 61 at the same time

2197: The difference between 659 times the last digit and the rest of the number is a multiple of 2197

2198: The number is a multiple of 2, 7 and 157 at the same time

2199: The number is a multiple of 3 and 733 at the same time

2200: The number is a multiple of 11 and 200 at the same time

[matias-2003]

The divisibility rules of every number from 1 to 100

1: Every number is a multiple of 1

2: The number ends in 0, 2, 4, 6 or 8

3: The sum of the digits is a multiple of 3

4: The last 2 digits are a multiple of 4

The 10s digit is even and the last digit is 0, 4 or 8

The 10s digit is odd and the last digit is 2 or 6

5: The number ends in 0 or 5

6: The number is a multiple of 2 and 3 at the same time

7: The difference between twice the last digit and the rest of the number is a multiple of 7

8: The last 3 digits are a multiple of 8

The 100s digit is even and the last 2 digits are a multiple of 8

The 100s digit is odd and the last 2 digits are 4 times an odd number

9: The sum of the digits is a multiple of 9

10: The number ends in 0

11: The difference between the last digit and the rest of the number is a multiple of 11

12: The number is a multiple of 3 and 4 at the same time

13: The sum of 4 times the last digit and the rest of the number is a multiple of 13

14: The number is a multiple of 2 and 7 at the same time

15: The number is a multiple of 3 and 5 at the same time

16: The last 4 digits are a multiple of 16

The 1,000s digit is even and the last 3 digits are a multiple of 16

The 1,000s digit is odd and the last 3 digits are 8 times an odd number

17: The difference between 5 times the last digit and the rest of the number is a multiple of 17

18: The number is a multiple of 2 and 9 at the same time

19: The sum of twice the last digit and the rest of the number is a multiple of 19

20: The number ends in 00, 20, 40, 60 or 80

21: The number is a multiple of 3 and 7 at the same time

22: The number is a multiple of 2 and 11 at the same time

23: The sum of 7 times the last digit and the rest of the number is a multiple of 23

24: The number is a multiple of 3 and 8 at the same time

25: The number ends in 00, 25, 50 or 75

26: The number is a multiple of 2 and 13 at the same time

27: The difference between 8 times the last digit and the rest of the number is a multiple of 27

28: The number is a multiple of 4 and 7 at the same time

29: The sum of thrice the last digit and the rest of the number is a multiple of 29

30: The number is a multiple of 3 and 10 at the same time

31: The difference between thrice the last digit and the rest of the number is a multiple of 31

32: The last 5 digits are a multiple of 32

The 10,000s digit is even and the last 4 digits are a multiple of 32

The 10,000s digit is odd and the last 4 digits are 16 times an odd number

33: The number is a multiple of 3 and 11 at the same time

34: The number is a multiple of 2 and 17 at the same time

35: The number is a multiple of 5 and 7 at the same time

36: The number is a multiple of 4 and 9 at the same time

37: The difference between 11 times the last digit and the rest of the number is a multiple of 37

38: The number is a multiple of 2 and 19 at the same time

39: The number is a multiple of 3 and 13 at the same time

40: The last 3 digits are a multiple of 40

The 100s digit is even and the last 2 digits are 00, 40 or 80

The 100s digit is odd and the last 2 digits are 20 or 60

41: The difference between 4 times the last digit and the rest of the number is a multiple of 41

42: The number is a multiple of 2, 3 and 7 at the same time

43: The sum of 13 times the last digit and the rest of the number is a multiple of 43

44: The number is a multiple of 4 and 11 at the same time

45: The number is a multiple of 5 and 9 at the same time

46: The number is a multiple of 2 and 23 at the same time

47: The difference between 14 times the last digit and the rest of the number is a multiple of 47

48: The number is a multiple of 3 and 16 at the same time

49: The sum of 5 times the last digit and the rest of the number is a multiple of 49

50: The number ends in 00 or 50

51: The number is a multiple of 3 and 17 at the same time

52: The number is a multiple of 4 and 13 at the same time

53: The sum of 16 times the last digit and the rest of the number is a multiple of 53

54: The number is a multiple of 2 and 27 at the same time

55: The number is a multiple of 5 and 11 at the same time

56: The number is a multiple of 7 and 8 at the same time

57: The number is a multiple of 3 and 19 at the same time

58: The number is a multiple of 2 and 29 at the same time

59: The sum of 6 times the last digit and the rest of the number is a multiple of 59

60: The number is a multiple of 3 and 20 at the same time

61: The difference between 6 times the last digit and the rest of the number is a multiple of 61

62: The number is a multiple of 2 and 31 at the same time

63: The number is a multiple of 7 and 9 at the same time

64: The last 6 digits are a multiple of 64

The 100,000s digit is even and the last 5 digits are a multiple of 64

The 100,000s digit is odd and the last 5 digits are 32 times an odd number

65: The number is a multiple of 5 and 13 at the same time

66: The number is a multiple of 2, 3 and 11 at the same time

67: The difference between 20 times the last digit and the rest of the number is a multiple of 67

68: The number is a multiple of 4 and 17 at the same time

69: The number is a multiple of 3 and 23 at the same time

70: The number is a multiple of 7 and 10 at the same time

71: The difference between 7 times the last digit and the rest of the number is a multiple of 71

72: The number is a multiple of 8 and 9 at the same time

73: The sum of 22 times the last digit and the rest of the number is a multiple of 73

74: The number is a multiple of 2 and 37 at the same time

75: The number is a multiple of 3 and 25 at the same time

76: The number is a multiple of 4 and 19 at the same time

77: The number is a multiple of 7 and 11 at the same time

78: The number is a multiple of 2, 3 and 13 at the same time

79: The sum of 8 times the last digit and the rest of the number is a multiple of 79

80: The last 4 digits are a multiple of 80

The 1,000s digit is even and the last 3 digits are a multiple of 80

The 1,000s digit is odd and the last 3 digits are 40 times an odd number

81: The difference between 8 times the last digit and the rest of the number is a multiple of 81

82: The number is a multiple of 2 and 41 at the same time

83: The sum of 25 times the last digit and the rest of the number is a multiple of 83

84: The number is a multiple of 3, 4 and 7 at the same time

85: The number is a multiple of 5 and 17 at the same time

86: The number is a multiple of 2 and 43 at the same time

87: The number is a multiple of 3 and 29 at the same time

88: The number is a multiple of 8 and 11 at the same time

89: The sum of 9 times the last digit and the rest of the number is a multiple of 89

90: The number is a multiple of 9 and 10 at the same time

91: The number is a multiple of 7 and 13 at the same time

92: The number is a multiple of 4 and 23 at the same time

93: The number is a multiple of 3 and 31 at the same time

94: The number is a multiple of 2 and 47 at the same time

95: The number is a multiple of 5 and 19 at the same time

96: The number is a multiple of 3 and 32 at the same time

97: The difference between 29 times the last digit and the rest of the number is a multiple of 97

98: The number is a multiple of 2 and 49 at the same time

99: The number is a multiple of 9 and 11 at the same time

100: The number ends in 00

[lilacchao]

matt parker just released an excellent video about number divisibility rules that has made me realise just how gorgeous divisibility rules are in base 6 (discussed under the cut)

every number in base 6 that doesn't end with 1 or 5 is divisible by 2 or 3, so to come up with a divisibility rule for some number 'n':

. divide 'n' by 10, 2, and 3 as much as possible (10 while it ends with 0, 2 while it ends with 2/4, 3 while it ends with 3) so it now ends with 1 or 5, call the result 'a', and call the product of all those 6s 2s and 3s 'b'

. (note that dividing by 2 or 3 is easy in base 6, just triple or double respectively and remove a zero - eg, 32/2 = 160/10 = 16, 15/3 = 30/10 = 3)

. split 'a' into two numbers, 'a1' (all but the last digit) and 'a2' (the last digit)

. if 'a2' is 1, the divisibility rule for 'a' is "take off the last digit of the number, multiply it by 'a1', and subtract it from the rest, and repeat until you get to a number small enough to recognise from multiplication tables"

. if 'a2' is 5, the divisibility rule for 'a' is "take off the last digit of the number, multiply it by 'a1', and add it to the rest, etc"

. split 'b' into two factors: all but the trailing zeroes, which is gonna be either 2^x or 3^x for some 'x', and 1 followed by all the trailing zeroes, which is equivalently 10^y for some 'y'

. given 2^x, define 'c' as 3^x, given 3^x, define 'c' as 2^x

. the divisibility rule for 'b' is to first multiply by 'c' then check if it ends with at least x+y zeroes

. the divisibility rule for 'n' is to do the divisibility rules for both 'a' and 'b', and if it's divisible by both 'a' and 'b', it's divisible by 'n'

example: is 11003535 divisible by 1220?

1220 = 41×20, so a=41, a1=4, a2=1, c=3, x=1, y=1

test for divisibility by 'b': 3×11003535 = 33015453, nope we wanted two zeroes and we found none, sad!

unnecessary now, but test for divisibility by 'a': 11003535 -> 1100353 - 4×5 = 1100321 -> 110032 - 4×1 = 110024 -> 11002 - 4×4 = 15534 -> 1553 - 4×4 = 1525 -> 152 - 4×5 = 120 -> 12, nope, 12 clearly isn't a multiple of 41

this is all a bit much to do in your head, but with basic familiarity with base 6, really easy on paper (or on phone like me just now!), and the divisibility rules are so easy to figure out!

[matias-2003]

The divisibility rules of every number from 801 to 900 with alternate methods for the composite numbers that end in 1, 3, 7 or 9

801: The difference between 80 times the last digit and the rest of the number is a multiple of 801

802: The number is a multiple of 2 and 401 at the same time

803: The sum of 241 times the last digit and the rest of the number is a multiple of 803

804: The number is a multiple of 4 and 201 at the same time

805: The number is a multiple of 5 and 161 at the same time

806: The number is a multiple of 2 and 403 at the same time

807: The difference between 242 times the last digit and the rest of the number is a multiple of 807

808: The number is a multiple of 8 and 101 at the same time

809: The sum of 81 times the last digit and the rest of the number is a multiple of 809

810: The number is a multiple of 10 and 81 at the same time

811: The difference between 81 times the last digit and the rest of the number is a multiple of 811

812: The number is a multiple of 4 and 203 at the same time

813: The sum of 244 times the last digit and the rest of the number is a multiple of 813

814: The number is a multiple of 2 and 407 at the same time

815: The number is a multiple of 5 and 163 at the same time

816: The number is a multiple of 16 and 51 at the same time

817: The difference between 245 times the last digit and the rest of the number is a multiple of 817

818: The number is a multiple of 2 and 409 at the same time

819: The sum of 82 times the last digit and the rest of the number is a multiple of 819

820: The number is a multiple of 20 and 41 at the same time

821: The difference between 82 times the last digit and the rest of the number is a multiple of 821

822: The number is a multiple of 2 and 411 at the same time

823: The sum of 247 times the last digit and the rest of the number is a multiple of 823

824: The number is a multiple of 8 and 103 at the same time

825: The number is a multiple of 25 and 33 at the same time

826: The number is a multiple of 2 and 413 at the same time

827: The difference between 248 times the last digit and the rest of the number is a multiple of 827

828: The number is a multiple of 4 and 207 at the same time

829: The sum of 83 times the last digit and the rest of the number is a multiple of 829

830: The number is a multiple of 10 and 83 at the same time

831: The difference between 83 times the last digit and the rest of the number is a multiple of 831

832: The number is a multiple of 13 and 64 at the same time

833: The sum of 250 times the last digit and the rest of the number is a multiple of 833

834: The number is a multiple of 2 and 417 at the same time

835: The number is a multiple of 5 and 167 at the same time

836: The number is a multiple of 4 and 209 at the same time

837: The difference between 251 times the last digit and the rest of the number is a multiple of 837

838: The number is a multiple of 2 and 419 at the same time

839: The sum of 84 times the last digit and the rest of the number is a multiple of 839

840: The number is a multiple of 21 and 40 at the same time

841: The difference between 84 times the last digit and the rest of the number is a multiple of 841

842: The number is a multiple of 2 and 421 at the same time

843: The sum of 253 times the last digit and the rest of the number is a multiple of 843

844: The number is a multiple of 4 and 211 at the same time

845: The number is a multiple of 5 and 169 at the same time

846: The number is a multiple of 2 and 423 at the same time

847: The difference between 254 times the last digit and the rest of the number is a multiple of 847

848: The number is a multiple of 16 and 53 at the same time

849: The sum of 85 times the last digit and the rest of the number is a multiple of 849

850: The number is a multiple of 17 and 50 at the same time

851: The difference between 85 times the last digit and the rest of the number is a multiple of 851

852: The number is a multiple of 4 and 213 at the same time

853: The sum of 256 times the last digit and the rest of the number is a multiple of 853

854: The number is a multiple of 2 and 427 at the same time

855: The number is a multiple of 5 and 171 at the same time

856: The number is a multiple of 8 and 107 at the same time

857: The difference between 257 times the last digit and the rest of the number is a multiple of 857

858: The number is a multiple of 2 and 429 at the same time

859: The sum of 86 times the last digit and the rest of the number is a multiple of 859

860: The number is a multiple of 20 and 43 at the same time

861: The difference between 86 times the last digit and the rest of the number is a multiple of 861

862: The number is a multiple of 2 and 431 at the same time so

863: The sum of 259 times the last digit and the rest of the number is a multiple of 863

864: The number is a multiple of 27 and 32 at the same time

865: The number is a multiple of 5 and 173 at the same time

866: The number is a multiple of 2 and 433 at the same time

867: The difference between 260 times the last digit and the rest of the number is a multiple of 867

868: The number is a multiple of 4 and 217 at the same time

869: The sum of 87 times the last digit and the rest of the number is a multiple of 869

870: The number is a multiple of 10 and 87 at the same time

871: The difference between 87 times the last digit and the rest of the number is a multiple of 871

872: The number is a multiple of 8 and 109 at the same time

873: The sum of 262 times the last digit and the rest of the number is a multiple of 873

874: The number is a multiple of 2 and 437 at the same time

875: The number is a multiple of 7 and 125 at the same time

876: The number is a multiple of 4 and 219 at the same time

877: The difference between 263 times the last digit and the rest of the number is a multiple of 877

878: The number is a multiple of 2 and 439 at the same time

879: The sum of 88 times the last digit and the rest of the number is a multiple of 879

880: The number is a multiple of 11 and 80 at the same time

881: The difference between 88 times the last digit and the rest of the number is a multiple of 881

882: The number is a multiple of 2 and 441 at the same time

883: The sum of 265 times the last digit and the rest of the number is a multiple of 883

884: The number is a multiple of 4 and 221 at the same time

885: The number is a multiple of 5 and 177 at the same time

886: The number is a multiple of 2 and 443 at the same time

887: The difference between 266 times the last digit and the rest of the number is a multiple of 887

888: The number is a multiple of 8 and 111 at the same time

889: The sum of 89 times the last digit and the rest of the number is a multiple of 889

890: The number is a multiple of 10 and 89 at the same time

891: The difference between 89 times the last digit and the rest of the number is a multiple of 891

892: The number is a multiple of 4 and 223 at the same time

893: The sum of 268 times the last digit and the rest of the number is a multiple of 893

894: The number is a multiple of 2 and 447 at the same time

895: The number is a multiple of 5 and 179 at the same time

896: The number is a multiple of 7 and 128 at the same time

897: The difference between 269 times the last digit and the rest of the number is a multiple of 897

898: The number is a multiple of 2 and 449 at the same time

899: The sum of 90 times the last digit and the rest of the number is a multiple of 899

900: The number is a multiple of 9 and 100 at the same time