Divisibility Rules

By LoneWolf271 on September 26, 2020 • ( 3 Comments )

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₎

Divisor	Divisibility Condition	Examples	Difficulty
1	No special condition. Any integer is divisible by 1.	2 is divisible by 1.	B
2	The last digit is even.	1294: 4 is even.	B
3	Sum 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
4	Twice the tens digit, plus the ones digit is divisible by 4.	832 = 3 * 2 + 2	E
5	The last digit is 0 or 5.	495: the last digit is 5.	B
6	It 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
7	Subtracting 2 times the last digit from the rest gives a multiple of 7.	483: 48 − (3 * 2) = 42 = 7 * 6	E
8	Add the last digit to twice the rest. The result must be divisible by 8.	56: (5 * 2) + 6 = 16.	E
9	Sum the digits. The result must be divisible by 9.	2880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9.	B
10	The last digit is 0.	130: the last digit is 0.	B
11	Add the digits in blocks of two from right to left. The result must be divisible by 11.	627: 6 + 27 = 33 = 3 * 11.	B
12	Subtract the last digit from twice the rest. The result must be divisible by 12.	324: 32 * 2 − 4 = 60 = 5 * 12.	E
13	Add 4 times the last digit to the rest. The result must be divisible by 13.	637: 63 + 7 * 4 = 91, 9 + 1 * 4 = 13	E
14	Add the last two digits to twice the rest. The result must be divisible by 14.	364: 3 * 2 + 64 = 70.	E
15	It is divisible by 3 and by 5.	390: it is divisible by 3 and by 5.	B
16	Add the last two digits to four times the rest. The result must be divisible by 16.	176: 1 * 4 + 76 = 80 = 16 * 5	E
17	Subtract 5 times the last digit from the rest.	221: 22 − 1 * 5 = 17.	E
18	It is divisible by 2 and by 9.	342: it is divisible by 2 and by 9.	B
19	Add twice the last digit to the rest.	437: 43 + 7 * 2 = 57.	E
20	It is divisible by 10, and the tens digit is even.	360: is divisible by 10, and 6 is even.	B
21	Subtracting twice the last digit from the rest gives a multiple of 21.	168: 16 − 8 * 2 = 0.	E
22	It is divisible by 2 and by 11.	352: it is divisible by 2 and by 11.	B
23	Add 7 times the last digit to the rest.	3128: 312 + 8 * 7 = 368. 36 + 8 × 7 = 92.	E
24	It is divisible by 3 and by 8.	552: it is divisible by 3 and by 8.	B
25	Examine the number formed by the last two digits.	134,250: 50 is divisible by 25.	E
26	Subtracting 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 * 8	I
27	Sum the digits in blocks of three from right to left.	2,644,272: 2 + 644 + 272 = 918.	E
28	It is divisible by 4 and by 7.	140: it is divisible by 4 and by 7.	B
29	Add three times the last digit to the rest.	348: 34 + 8 * 3 = 58.	E
30	It is divisible by 3 and by 10.	270: it is divisible by 3 and by 10.	B
31	Subtract three times the last digit from the rest.	837: 83 − 3 * 7 = 62	E
32	Add the last two digits to 4 times the rest.	1312: (13 * 4) + 12 = 64.	E
33	Add 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

Divisor	Divisibility Condition	Examples¹	Difficulty
35	Number must be divisible by 7 ending in 0 or 5.		B
37	Subtract 11 times the last digit from the rest.	925: 92 − (5 * 11) = 37.	E
39	Add 4 times the last digit to the rest.	351: 35 + (1 * 4) = 39	E
41	Subtract 4 times the last digit from the rest.	738: 73 − 8 * 4 = 41.	E
43	Add 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
45	The number must be divisible by 9 ending in 0 or 5.	2025: Ends in 5 and 2 + 0 + 2 + 5 = 9.	E
47	Subtract 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
49	Add 5 times the last digit to the rest.	1,127: 112 + (7 * 5) = 147. 147: 14 + (7 * 5) = 49	E
50	The last two digits are 00 or 50.	134,250: 50.	B
51	Subtract 5 times the last digit from the rest.	204: 20 – (4 * 5) = 0	E
53	Add 16 times the last digit to the rest.	3657: 365 + (7 * 16) = 477 = 9 * 53	E
55	Number must be divisible by 11 ending in 0 or 5.		E
57	Subtract 17 times the last digit from the rest.	3591: 359 − 17 = 342, 34 − 2 * 17 = 0.	E
59	Add 6 times the last digit to the rest.	295: 29 + 5 * 6= 59	E
61	Subtract 6 times the last digit from the rest.	732: 73 – (2 * 6) = 61	E
64²	The number formed by the last six digits must be divisible by 64.	2,640,000 is divisible by 64.	I
65	Number must be divisible by 13 ending in 0 or 5.		B
67	Subtract 20 times the last digit from the rest.	4489: 448 – 9 * 20 = 448 – 180 = 268.	E
69	Add 7 times the last digit to the rest.	345: 34 + 5 * 7 = 69	E
71	Subtract 7 times the last digit from the rest.	852: 85 – (2 * 7) = 71	E
73	Add 22 times the last digit from the rest.	5329: 532 + 22 * 9 = 730, 7 + 22 * 3 = 73.	E
75	Number must be divisible by 3 ending in 00, 25, 50 or 75.		E
77	Form the alternating sum of blocks of three from right to left.	76,923: 923 – 76 = 847.	I
79	Add 8 times the last digit to the rest.	711: 71 + 1 * 8 = 79	E
81	Subtract 8 times the last digit from the rest.	162: 16 – (2 * 8) = 0	E
83	Add 25 times the last digit to the rest.	581: 58 + (1 * 25) = 83	E
85	Number must be divisible by 17 ending in 0 or 5.	30,855: 3085 – 25 = 3060 = 17 * 18. And the number ends in 5.	E
87	Subtract 26 times the last digit from the rest.	15138: 1513 − 8 * 26 = 1305, 130 − 5 * 26 = 0.	E
89	Add 9 times the last digit to the rest.	801: 80 + 1 * 9 = 89	E
91	Subtract 9 times the last digit from the rest.	182: 18 – (2 * 9) = 0	E
95	Number 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
97	Add the last two digits to 3 times the rest.	485: (3 * 4) + 85 = 97	E – I
99	Add the digits in blocks of two from right to left.	144,837: 14 + 48 + 37 = 99.	E
100	Ends with at least two zeros.	14100: It has two zeros at the end.	B
101	Form the alternating sum of blocks of two from right to left.	40,299: 4 – 2 + 99 = 101.	I
103	Subtract the last two digits from 3 times the rest.	5356: (53 * 3) – 56 = 103	E – I
107	Subtract the last two digits from 7 times the rest.	1712: 17 * 7 – 12 = 107	I – H
109	Add 11 times the last digit to the rest.	654: 65 + (11 * 4) = 109	E
111	Add the digits in blocks of three from right to left.	1,370,184: 1 + 370 + 184 = 555	E
113	Add 34 times the last digit from the rest.	3842: 384 + 34 * 2 = 452, 45 + 34 * 2 = 113.	I
121	Subtract 12 times the last digit from the rest.	847: 84 – 12 * 7 = 0	E
125	The number formed by the last three digits must be 000, 125, 250, 375, 500, 625, 750 and 875.	2125 is divisible by 125.	I
127	Subtract 38 times the last digit from the rest.	4953: 495 – 38 * 3 = 381, 38 – 38 * 1 = 0.	I
128³	The number formed by the last seven digits must be divisible by 128.	11,280,000 is divisible by 128.	H
131	Subtract 13 times the last digit from the rest.	1834: 183 – 13 * 4 = 131, 13 – 1 * 13 = 0.	E
137	Form the alternating sum of blocks of three from right to left.	340,171: 171 – 34 = 137.	I
139	Add 14 times the last digit from the rest.	1946: 194 + 14 * 6 = 278, 27 + 14 * 8 = 139.	E
143	Add 43 times the last digit to the rest.	6149: 614 + 43 * 9 = 1001, 100 + 1 * 43 = 143.	I
149	Add 15 times the last digit from the rest.	2235: 223 + 15 * 5 = 298, 29 + 15 * 8 = 149.	E
151	Subtract 15 times the last digit from the rest.	66,893: 6689 – 15 * 3 = 6644 = 151 * 44.	E
157	Subtract 47 times the last digit from the rest.	7536: 753 – 47 * 6 = 471, 47 – 47 = 0.	I
163	Add 49 times the last digit to the rest.	26,569: 2656 + 441 = 3097 = 163 * 19.	I
167	Subtract 5 times the last two digits from the rest.	53,774: 537 – 5 * 74 = 167.	E
173	Add 52 times the last digit to the rest.	8996: 899 + 52 * 6 = 1211, 121 + 52 = 173.	H
179	Add 18 times the last digit to the rest.	3222: 322 + 18 * 2 = 358, 35 + 18 * 8 = 179.	E
181	Subtract 18 times the last digit to the rest.	3258: 325 – 18 * 8 = 181, 18 – 18 = 0.	E
191	Subtract 19 times the last digit to the rest.	3629: 362 – 19 * 9 = 191, 19 – 1 * 19 = 0.	E
193	Add 58 times the last digit to the rest.	11194: 1119 + 58 * 4 = 1351, 135 + 58 = 193.	H
197	Subtract 59 times the last digit to the rest.	11820: 118 – 59 * 2 = 0.	H
199	Add 20 times the last digit to the rest.	3980: 39 + 20 * 8 = 199.	I
200	Last 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
221	Subtract 21 times the last digit to the rest.	44521: 4452 – 21 * 1 = 4431, 443 – 21 * 1 = 422, 42 – 21 * 2 = 0.	E
223	Add 67 times the last digit to the rest.	49729: 4972 + 67 * 9 = 5575, 557 + 67 * 5 = 892, 89 + 67 * 2 = 223.	H
225	Last 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
227	Subtract 68 times the last digit to the rest.	51756: 5175 – 68 * 6 = 4767, 476 – 68 * 7 = 0.	H
229	Add 23 times the last digit to the rest.	52441: 5244 + 23 * 1 = 5267, 526 + 23 * 7 = 687, 68 + 23 * 7 = 229.	E
233	Add 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
239	Add 24 times the last digit to the rest.	57121: 5712 + 24 * 1 = 5736, 573 + 24 * 6 = 717, 71 + 24 * 7 = 239.	E
241	Subtract 24 times the last digit to the rest.	58081: 5808 – 24 * 1 = 5784, 578 – 24 * 4 = 482, 48 – 24 * 2 = 0.	E
250	The number formed by the last three digits must be 000, 250, 500 and 750.	1,327,750 is divisible by 250.	I
251	Subtract 25 times the last digit to the rest.	63001: 6300 – 25 * 1 = 6275, 627 – 25 * 5 = 502, 50 – 25 * 2 = 0.	E
256⁴	The number formed by the last eight digits must be divisible by 256.	225,600,000 is divisible by 256.	H
257	Subtract 77 times the last digit to the rest.	66049: 6604 – 77 * 9 = 5911, 591 – 77 * 1 = 514 = 257 * 2.	S
263	Add 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
269	Add 27 times the last digit to the rest.	72361: 7236 + 27 * 1 = 7263, 726 + 27 * 3 = 807, 80 + 27 * 7 = 269.	E
271	Subtract 27 times the last digit from the rest.	73441: 7344 – 27 * 1 = 7317, 731 – 27 * 7 = 542, 54 – 27 * 2 = 0.	E
277	Subtract 83 times the last digit from the rest.	76729: 7672 – 83 * 9 = 6925, 692 – 83 * 5 = 277.	S
281	Subtract 28 times the last digit from the rest.	78961: 7896 – 28 * 1 = 7868, 786 – 28 × 8 = 562, 56 – 28 2 = 0.	E
283	Add 85 times the last digit to the rest.	80089: 8008 + 85 * 9 = 8773, 877 + 85 * 3 = 1132, 113 + 85 * 2 = 283.	S
293	Add 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
300	Last 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
329	Add 33 times the last digit to the rest.	9541: 954 + 1 * 33=987. 987=3 * 329.	I
331	Subtract 33 times the last digit from the rest.	22177: 2217-7 * 231=1986. 1986=6 * 331.	I
333	Add the digits in blocks of three from right to left.	410,922: 410 + 922 = 1,332	E
369	Add 37 times the last digit to the rest.	8487: 848 + 7 * 37 = 1107.	I
375	The 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
499	Add the last three digits to two times the rest.	74,351: 74 * 2 + 351 = 499.	E – H
500	Ends with 000 or 500.	47,500 is divisible by 500.	B
512⁵	The number formed by the last nine digits must be divisible by 512.	1,512,000,000 is divisible by 512.	H
625	Ends in 0000, 0625, 1250, 1875, 2500, 3125, 3750, 4375, 5000, 5625, 6250, 6875, 7500, 8125, 8750 or 9375.	567,886,875: 6875.	H
983	Add the last three digits to seventeen times the rest.	64878: 64 * 17 + 878=1966. 1966=2 * 983	S
987	Add the last three digits to thirteen times the rest.	30597: 30 * 13 + 597=987	S
989	Add the last three digits to eleven times the rest.	21758: 21 * 11 + 758 = 989	H – S
993	Divisible 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
997	Add the last three digits to three times the rest.	157,526: 157 * 3 + 526= 997	E – H
999	Add the digits in blocks of three from right to left.	235,764: 235 + 764 = 999	E
1000	Ends with at least three zeros.	2000 ends with 3 zeros.	B
1024^{6, 7}	The number formed by the last ten digits must be divisible by 1024.	107374063616: 7374063616 is divisible by 1024	S

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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Categories: Challenges, Random Thoughts

Tagged as: Divisibility Rules, Mathematics

3 replies »

Wu Ming 無明 says:

September 27, 2020 at 12:04 am

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 ) 😨😨😨

LikeLiked by 1 person

William says:

September 30, 2020 at 12:25 pm

你冇嘢啊嘛，我淨係識得LMF

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- mynewchapterinlife says:
  
  November 2, 2020 at 1:40 am
  
  我冇嘢呀，呢篇係出自我家小老闆手筆，冇法啦😬
  
  LMF 好嘢嚟吖 👍👍
  
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