The Divisibility Rules You Didn't Learn In School

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Divisibility rules are some of the most helpful tricks you can learn in math. To be able to know if one number divides another just by looking at it is both incredibly helpful and easy, and while you’ve likely heard of some of the most common rules, there are some incredibly simple ones that you probably never learned. In this video, I explain not only how to find these rules, but also how to calculate them easily in your head

Divisibility rules:

Note: B represents the last digit of a number and A represents all other digits.

Negative numbers: Use the divisibility of its positive counterpart.

0: No number is divisible by 0.

1: All integers are divisible by 1.

2: The last digit must be divisible by 2.

3: The digits must sum to a multiple of 3.

4: The last two digits must be divisible by 4.

5: The last digit must be divisible by 5.

6: Divisible by 2 and 3.

7: A+5B must be divisible by 7.

8: The last three digits must be divisible by 8.

9: The digits must sum to a multiple of 9.

10: The last digit must be a multiple of 10.

11: The alternating sum of the digits must be divisible by 11.

12: Divisible by 3 and 4.

13: A+4B must be divisible by 13.

14: Divisible by 2 and 7.

15: Divisible by 3 and 5.

16: The last four digits must be divisible by 16.

17: A-5B must be divisible by 17.

18: Divisible by 2 and 9.

19: A+2B must be divisible by 19.

20: Divisible by 4 and 5.

21: Divisible by 3 and 7.

22: Divisible by 2 and 11.

23: A+7B must be divisible by 23.

24: Divisible by 3 and 8.

25: The last two digits must be a multiple of 25.

26: Divisible by 2 and 13.
27: A-8B must be divisible by 27.
28: Divisible by 2 and 14.
29: A+3B must be divisible by 29.
30: Divisible by 3 and 10.
31: A-3B must be divisible by 31.
32: The last five digits must be divisible by 32.
33: Divisible by 3 and 11.
34: Divisible by 2 and 17.
35: Divisible by 5 and 7.
36: Divisible by 4 and 9.
37: A-11B must be divisible by 37.
38: Divisible by 2 and 19.
39: Divisible by 3 and 13.
40: Divisible by 5 and 8.
41: A-4B must be divisible by 41.
42: Divisible by 2 and 3 and 7.
43: A+13B must be divisible by 43.
44: Divisible by 4 and 11.
45: Divisible by 5 and 9.
46: Divisible by 2 and 23.
47: A-14B must be divisible by 47.
48: Divisible by 3 and 16.
49: A+5B must be divisible by 49.
50: The last two digits must be divisible by 50.
51: Divisible by 3 and 17.
52: Divisible by 4 and 13.
53: A+16B must be divisible by 53.
54: Divisible by 2 and 27.
55: Divisible by 5 and 11.
56: Divisible by 7 and 8.
57: Divisible by 3 and 19.
58: Divisible by 2 and 29.
59: A+6B must be divisible by 59.
60: Divisible by 3 and 4 and 5.
61: A-6B must be divisible by 61.
62: Divisible by 2 and 31.
63: Divisible by 7 and 9.
64: The last six digits must be divisible by 64.
65: Divisible by 5 and 13.
66: Divisible by 6 and 11.
67: A-20B must be divisible by 67.
68: Divisible by 4 and 17.
69: Divisible by 3 and 23.
70: Divisible by 7 and 10.
71: A-7B must be divisible by 71.
72: Divisible by 8 and 9.
73: A-22B must be divisible by 73.
74: Divisible by 2 and 37.
75: Divisible by 3 and 25.
76: Divisible by 4 and 19.
77: Divisible by 7 and 11.
78: Divisible by 2 and 3 and 13.
79: A+8B must be divisible by 79.
80: Divisible by 5 and 16.
81: A-8B must be divisible by 81.
82: Divisible by 2 and 41.
83: A+25B is divisible by 83.
84: Divisible by 3 and 4 and 7.
85: Divisible by 5 and 17.
86: Divisible by 2 and 43.
87: Divisible by 3 and 29.
88: Divisible by 8 and 11.
89: A+9B must be divisible by 89.
90: Divisible by 9 and 10.
91: Divisible by 7 and 13.
92: Divisible by 4 and 23.
93: Divisible by 3 and 31.
94: Divisible by 2 and 47.
95: Divisible by 5 and 19.
96: Divisible by 3 and 32.
97: A-29B must be divisible by 97.
98: Divisible by 2 and 49.
99: Divisible by 9 and 11.
100: The last two digits must be divisible by 100.

Greater than 100:

Prime Numbers: For larger primes, B will increase accordingly in length. Multiply your 100...A+B expression by the smallest whole number that yields a coefficient of A that is one away from the prime. Factor out that prime multiple of A and get rid of it, the resulting expression is your rule. If left with -A, multiply by -1 to yield an easier formula.

Composite Numbers: Check all prime power factors.