Divisibility Rules

Hello guys! It has been a long time since my last blog!

Today I’m going to share some interesting rules in Mathematics that is about “How to determine whether a given integer is divisible by a fixed divisor without actually performing the division”.

Hello guys! It has been a long time since my last blog!

Today I’m going to share some interesting rules in Mathematics that is about “How to determine whether a given integer is divisible by a fixed divisor without actually performing the division“.

Read on for more information!


As you could guess, these rules are called “Divisibility Rules“.

It is something convenient that you may need to use from Primary 4 to Secondary 1. For example, in finding L.C.M. and H.C.F. In fact, there is a section in my Mathematics book that even teaches the divisibility of 2, 3, 4, 5, 6, 8, 9, 10 and 11!

As we get older and learn more Mathematics, maybe we need to know the divisibility of 1 to 30!

Oops … , but don’t be afraid!

Let me show you the divisibility rules for 1 to 33 in the table below first and I will also include other information such as examples and difficulties.

Well, I don’t think you will finish reading my post without falling asleep… 😛 because the chart is so long that I may need 30 years to finish it!

Remarks:

If the divisor has multiple divisibility conditions, only the easiest one will be shown

Difficulty: B=Beginner, E=Easy, I=Intermediate, H=Hard, S=Expert

The following tables are extracted from wikipedia. (https://en.wikipedia.org/wiki/Divisibility_rule#cite_note-Pascal’s-criterion-2)

DivisorDivisibility ConditionExamplesDifficulty
1No special condition. Any integer is divisible by 1.2 is divisible by 1.B
2The last digit is even.1294: 4 is even.B
3Sum the digits. The result must be divisible by 3.405 → 4 + 0 + 5 = 9 and 636 → 6 + 3 + 6 = 15 which both are clearly divisible by 3.
B
4Twice the tens digit, plus the ones digit is divisible by 4.832 = 3 * 2 + 2E
5The last digit is 0 or 5.495: the last digit is 5.B
6It is divisible by 2 and by 3.1458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, so the number is divisible by 6.B
7Subtracting 2 times the last digit from the rest gives a multiple of 7.483: 48 − (3 * 2) = 42 = 7 * 6E
8Add the last digit to twice the rest. The result must be divisible by 8.56: (5 * 2) + 6 = 16.E
9Sum the digits. The result must be divisible by 9.2880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9.B
10The last digit is 0.130: the last digit is 0.B
11Add the digits in blocks of two from right to left. The result must be divisible by 11.627: 6 + 27 = 33 = 3 * 11.B
12Subtract the last digit from twice the rest. The result must be divisible by 12.324: 32 * 2 − 4 = 60 = 5 * 12.E
13Add 4 times the last digit to the rest. The result must be divisible by 13.637: 63 + 7 * 4 = 91, 9 + 1 * 4 = 13E
14Add the last two digits to twice the rest. The result must be divisible by 14.364: 3 * 2 + 64 = 70.E
15It is divisible by 3 and by 5.390: it is divisible by 3 and by 5.B
16Add the last two digits to four times the rest. The result must be divisible by 16.176: 1 * 4 + 76 = 80 = 16 * 5E
17Subtract 5 times the last digit from the rest.221: 22 − 1 * 5 = 17.E
18It is divisible by 2 and by 9.342: it is divisible by 2 and by 9.B
19Add twice the last digit to the rest.437: 43 + 7 * 2 = 57.E
20It is divisible by 10, and the tens digit is even.360: is divisible by 10, and 6 is even.B
21Subtracting twice the last digit from the rest gives a multiple of 21.168: 16 − 8 * 2 = 0.E
22It is divisible by 2 and by 11.352: it is divisible by 2 and by 11.B
23Add 7 times the last digit to the rest.3128: 312 + 8 * 7 = 368. 36 + 8 × 7 = 92.E
24It is divisible by 3 and by 8.552: it is divisible by 3 and by 8.B
25Examine the number formed by the last two digits. 134,250: 50 is divisible by 25.E
26Subtracting 5 times the last digit from 2 times the rest of the number gives a multiple of 26.1248 : (124 * 2) – (8 * 5) = 208 = 26 * 8I
27Sum the digits in blocks of three from right to left.2,644,272: 2 + 644 + 272 = 918.E
28It is divisible by 4 and by 7.140: it is divisible by 4 and by 7.B
29Add three times the last digit to the rest.348: 34 + 8 * 3 = 58.E
30It is divisible by 3 and by 10.270: it is divisible by 3 and by 10.B
31Subtract three times the last digit from the rest.837: 83 − 3 * 7 = 62E
32Add the last two digits to 4 times the rest.1312: (13 * 4) + 12 = 64.E
33Add the digits in blocks of two from right to left.2145: 21 + 45 = 66.B

In addition, I will provide rules for some more notable divisors beyond 33.

These are:

35, 37, 39, 41, 43, 45, 47, 49 – 51, 53, 55, 57, 59, 61, 64 – 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99 – 101, 103, 107, 109, 111, 113, 121, 125, 127 – 128, 131, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199 – 200, 211, 223, 225, 227, 229, 233, 239, 241, 250 – 251, 256 – 257, 263 ,269, 271, 277, 281, 283, 293, 300, 329, 331, 333, 369, 375, 499 – 500, 512, 625, 983, 987, 989, 993, 997, 999 – 1000 and 1024

DivisorDivisibility ConditionExamples1Difficulty
35Number must be divisible by 7 ending in 0 or 5.B
37Subtract 11 times the last digit from the rest.925: 92 − (5 * 11) = 37.E
39Add 4 times the last digit to the rest.351: 35 + (1 * 4) = 39E
41Subtract 4 times the last digit from the rest.738: 73 − 8 * 4 = 41.E
43Add 13 times the last digit to the rest.36,249: 3624 + 9 * 13 = 3741, 
374 + 1 * 13 = 387, 
38 + 7 * 13 = 129, 
12 + 9 * 13 = 129 = 43 * 3.
E
45The number must be divisible by 9 ending in 0 or 5.2025: Ends in 5 and 2 + 0 + 2 + 5 = 9.E
47Subtract 14 times the last digit from the rest.1,642,979: 164297 − 9 * 14 = 164171, 
16417 − 14 = 16403, 
1640 − 3 * 14 = 1598, 
159 − 8 * 14 = 47.
E
49Add 5 times the last digit to the rest.1,127: 112 + (7 * 5) = 147.
147: 14 + (7 * 5) = 49
E
50The last two digits are 00 or 50.134,250: 50.B
51Subtract 5 times the last digit from the rest.204: 20 – (4 * 5) = 0E
53Add 16 times the last digit to the rest.3657: 365 + (7 * 16) = 477 = 9 * 53E
55Number must be divisible by 11 ending in 0 or 5.E
57Subtract 17 times the last digit from the rest.3591: 359 − 17 = 342, 
34 − 2 * 17 = 0.
E
59Add 6 times the last digit to the rest.295: 29 + 5 * 6= 59E
61Subtract 6 times the last digit from the rest.732: 73 – (2 * 6) = 61E
642The number formed by the last six digits must be divisible by 64.2,640,000 is divisible by 64.I
65Number must be divisible by 13 ending in 0 or 5.B
67Subtract 20 times the last digit from the rest.4489: 448 – 9 * 20 = 448 – 180 = 268.E
69Add 7 times the last digit to the rest.345: 34 + 5 * 7 = 69E
71Subtract 7 times the last digit from the rest.852: 85 – (2 * 7) = 71E
73Add 22 times the last digit from the rest.5329: 532 + 22 * 9 = 730, 
7 + 22 * 3 = 73.
E
75Number must be divisible by 3 ending in 00, 25, 50 or 75.E
77Form the alternating sum of blocks of three from right to left.76,923: 923 – 76 = 847.I
79Add 8 times the last digit to the rest.711: 71 + 1 * 8 = 79E
81Subtract 8 times the last digit from the rest.162: 16 – (2 * 8) = 0E
83Add 25 times the last digit to the rest.581: 58 + (1 * 25) = 83E
85Number must be divisible by 17 ending in 0 or 5.30,855: 3085 – 25 = 3060 = 17 * 18. And the number ends in 5.E
87Subtract 26 times the last digit from the rest.15138: 1513 − 8 * 26 = 1305, 
130 − 5 * 26 = 0.
E
89Add 9 times the last digit to the rest.801: 80 + 1 * 9 = 89E
91Subtract 9 times the last digit from the rest.182: 18 – (2 * 9) = 0E
95Number must be divisible by 19 ending in 0 or 5.51,585: 5158 + 10 = 5168, 
516 + 16 = 532, 
53 + 4 = 57 = 19 * 3. And the number ends in 5.
E
97Add the last two digits to 3 times the rest.485: (3 * 4) + 85 = 97E – I
99Add the digits in blocks of two from right to left.144,837: 14 + 48 + 37 = 99.E
100Ends with at least two zeros.14100: It has two zeros at the end.B
101Form the alternating sum of blocks of two from right to left.40,299: 4 – 2 + 99 = 101.I
103Subtract the last two digits from 3 times the rest.5356: (53 * 3) – 56 = 103E – I
107Subtract the last two digits from 7 times the rest.1712: 17 * 7 – 12 = 107I – H
109Add 11 times the last digit to the rest.654: 65 + (11 * 4) = 109E
111Add the digits in blocks of three from right to left.1,370,184: 1 + 370 + 184 = 555E
113Add 34 times the last digit from the rest.3842: 384 + 34 * 2 = 452, 
45 + 34 * 2 = 113.
I
121Subtract 12 times the last digit from the rest.847: 84 – 12 * 7 = 0E
125The number formed by the last three digits must be 000, 125, 250, 375, 500, 625, 750 and 875.2125 is divisible by 125.I
127Subtract 38 times the last digit from the rest.4953: 495 – 38 * 3 = 381, 
38 – 38 * 1 = 0.
I
1283The number formed by the last seven digits must be divisible by 128.11,280,000 is divisible by 128.H
131Subtract 13 times the last digit from the rest.1834: 183 – 13 * 4 = 131, 
13 – 1 * 13 = 0.
E
137Form the alternating sum of blocks of three from right to left.340,171: 171 – 34 = 137.I
139Add 14 times the last digit from the rest.1946: 194 + 14 * 6 = 278, 
27 + 14 * 8 = 139.
E
143Add 43 times the last digit to the rest.6149: 614 + 43 * 9 = 1001, 
100 + 1 * 43 = 143.
I
149Add 15 times the last digit from the rest.2235: 223 + 15 * 5 = 298, 
29 + 15 * 8 = 149.
E
151Subtract 15 times the last digit from the rest.66,893: 6689 – 15 * 3 = 6644 = 151 * 44.E
157Subtract 47 times the last digit from the rest.7536: 753 – 47 * 6 = 471, 
47 – 47 = 0.
I
163Add 49 times the last digit to the rest.26,569: 2656 + 441 = 3097 = 163 * 19.I
167Subtract 5 times the last two digits from the rest.53,774: 537 – 5 * 74 = 167.E
173Add 52 times the last digit to the rest.8996: 899 + 52 * 6 = 1211, 
121 + 52 = 173.
H
179Add 18 times the last digit to the rest.3222: 322 + 18 * 2 = 358, 
35 + 18 * 8 = 179.
E
181Subtract 18 times the last digit to the rest.3258: 325 – 18 * 8 = 181, 
18 – 18 = 0.
E
191Subtract 19 times the last digit to the rest.3629: 362 – 19 * 9 = 191, 
19 – 1 * 19 = 0.
E
193Add 58 times the last digit to the rest.11194: 1119 + 58 * 4 = 1351, 
135 + 58 = 193.
H
197Subtract 59 times the last digit to the rest.11820: 118 – 59 * 2 = 0.H
199Add 20 times the last digit to the rest.3980: 39 + 20 * 8 = 199.I
200Last three digits of the number are 000, 200, 400, 600 and 800.34,400: The third last digit is 4, and the last two digits are zeroes.E
221Subtract 21 times the last digit to the rest.44521: 4452 – 21 * 1 = 4431, 
443 – 21 * 1 = 422, 
42 – 21 * 2 = 0.
E
223Add 67 times the last digit to the rest.49729: 4972 + 67 * 9 = 5575, 
557 + 67 * 5 = 892, 
89 + 67 * 2 = 223.
H
225Last two digits of the number are “00”, “25”, “50”, or “75” and the sum of the digits is a multiple of 9.15,075: 75 is at the end and 1 + 5 + 0 + 7 + 5 = 18 = 2 * 9.I
227Subtract 68 times the last digit to the rest.51756: 5175 – 68 * 6 = 4767, 
476 – 68 * 7 = 0.
H
229Add 23 times the last digit to the rest.52441: 5244 + 23 * 1 = 5267, 
526 + 23 * 7 = 687, 
68 + 23 * 7 = 229.
E
233Add 70 times the last digit to the rest.54289: 5428 + 70 * 9 = 6058, 
605 + 70 * 8 = 1165, 
116 + 70 * 5 = 466, 
46 + 70 * 6 = 466 = 233 * 2.
S
239Add 24 times the last digit to the rest.57121: 5712 + 24 * 1 = 5736, 
573 + 24 * 6 = 717, 
71 + 24 * 7 = 239.
E
241Subtract 24 times the last digit to the rest.58081: 5808 – 24 * 1 = 5784, 
578 – 24 * 4 = 482, 
48 – 24 * 2 = 0.
E
250The number formed by the last three digits must be 000, 250, 500 and 750.1,327,750 is divisible by 250.I
251Subtract 25 times the last digit to the rest.63001: 6300 – 25 * 1 = 6275, 
627 – 25 * 5 = 502, 
50 – 25 * 2 = 0.
E
2564The number formed by the last eight digits must be divisible by 256.225,600,000 is divisible by 256.H
257Subtract 77 times the last digit to the rest.66049: 6604 – 77 * 9 = 5911, 
591 – 77 * 1 = 514 = 257 * 2.
S
263Add 79 times the last digit to the rest.69169: 6916 + 79 * 9 = 7627, 
762 + 79 * 7 = 1315, 
131 + 79 * 5 = 526, 
52 + 79 * 6 = 526 = 263 * 2.
S
269Add 27 times the last digit to the rest.72361: 7236 + 27 * 1 = 7263, 
726 + 27 * 3 = 807, 
80 + 27 * 7 = 269.
E
271Subtract 27 times the last digit from the rest.73441: 7344 – 27 * 1 = 7317, 
731 – 27 * 7 = 542, 
54 – 27 * 2 = 0.
E
277Subtract 83 times the last digit from the rest.76729: 7672 – 83 * 9 = 6925, 
692 – 83 * 5 = 277.
S
281Subtract 28 times the last digit from the rest.78961: 7896 – 28 * 1 = 7868, 
786 – 28 *× 8 = 562, 
56 – 28 * 2 = 0.
E
283Add 85 times the last digit to the rest.80089: 8008 + 85 * 9 = 8773, 
877 + 85 * 3 = 1132, 
113 + 85 * 2 = 283.
S
293Add 88 times the last digit to the rest.85849: 8584 + 88 * 9 = 9376, 
937 + 88 * 6 = 1465, 
146 + 88 * 5 = 586, 
58 + 88 * 6 = 586 = 293 * 2.
S
300Last two digits of the number are “00”, and the result of sum the digits must be divisible by 3.3,300: The result of sum the digits is 6, and the last two digits are zeroes.S
329Add 33 times the last digit to the rest.9541: 954 + 1 * 33=987. 987=3 * 329.I
331Subtract 33 times the last digit from the rest.22177: 2217-7 * 231=1986. 1986=6 * 331.I
333Add the digits in blocks of three from right to left.410,922: 410 + 922 = 1,332E
369Add 37 times the last digit to the rest.8487: 848 + 7 * 37 = 1107.I
375The number formed by the last 3 digits must be divisible by 125 and the sum of all digits is a multiple of 3.140,625: 625 = 125 * 5;
1 + 4 + 0 + 6 + 2 + 5 = 18 = 6 * 3.
H
499Add the last three digits to two times the rest.74,351: 74 * 2 + 351 = 499.E – H
500Ends with 000 or 500.47,500 is divisible by 500.B
5125The number formed by the last nine digits must be divisible by 512.1,512,000,000 is divisible by 512.H
625Ends in 0000, 0625, 1250, 1875, 2500, 3125, 3750, 4375, 5000, 5625, 6250, 6875, 7500, 8125, 8750 or 9375.567,886,875: 6875.H
983Add the last three digits to seventeen times the rest.64878: 64 * 17 + 878=1966. 1966=2 * 983S
987Add the last three digits to thirteen times the rest.30597: 30 * 13 + 597=987S
989Add the last three digits to eleven times the rest.21758: 21 * 11 + 758 = 989H – S
993Divisible by 331 and by 3.8937: 8 + 7 = 15 (3 and 9 is omitted because they are divisible by 3)
15 = 3 * 5
8937: 893 – 7 * 33 = 662 = 331 * 2
I
997Add the last three digits to three times the rest.157,526: 157 * 3 + 526= 997E – H
999Add the digits in blocks of three from right to left.235,764: 235 + 764 = 999E
1000Ends with at least three zeros.2000 ends with 3 zeros.B
10246, 7The number formed by the last ten digits must be divisible by 1024.107374063616: 7374063616 is divisible by 1024S

1 With the exception of 3 and 9, the zero is omitted if the divisibility process contains zero (e.g. see divisor 101 and 137)
2 This divisibility condition only applies to a 6-digit number or more only.
3 This divisibility condition only applies to a 7-digit number or more only.
4 This divisibility condition only applies to a 8-digit number or more only.
5 This divisibility condition only applies to a 9-digit number or more only.
6 This divisibility condition only applies to a 10-digit number or more only.
7 210 = 1024 which the number formed by the last 10 digits must be divisible by 1024.

In fact, the last divisibility rule can be generalised as below:

A number y is divisible by 2n, if the number x formed by its last n digits is divisible by 2n, where n is any integer from 1 to ∞.

For example, the number 1262144 is divisible by 64 (26) as the number formed by the last six digits (262144) is divisible by 64.

Enjoy!


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  1. Dear lonewolf271, welcome back to blogging world, Hil Hil e e 😯 tries very hard to digest and attempt to comment ~ unfortunately I failed to make any sense of it. 😱😱😱

    With hindsight, I am so glad that I managed to get my 天下無敵E in Math (public exam) , if I were born in this era, I would be …. (lost for words ) 😨😨😨

    Liked by 1 person

    Reply

  2. 你冇嘢啊嘛,我淨係識得LMF

    Like

    Reply

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